H\«tt«r*«itg   fff 


Name  of  Book  and  Volume, 


Division 

Range 

Shelf.... 

Received. &.&b&..       187^ 


University  of  California. 


G-IFX   OF 


E     TMENTS'  OF  MECHANICS. 


THE 


ELEMENTS  OF   MECHANICS 


JAMES    RENWICK,    LL.D., 


PROFESSOR  OF  NATURAL  EXPERIMENTAL'  PHILOSOPHY 
AND  CHEMISTRY 


C  O  L.-U  M  B  I  A    COLLEGE, 

NEW-YORK. 


CAREY    &    LEA,    C  H  E  S  N  U  T-S  T  R  E  E  T. 

1832. 


Entered  according  to  the  Act  of  Congress,  in  the  year  one  thousand  eight  hun- 
dred and  thirty-twa,  by  James  Renwick,  in  the  Clerk's  Office  of  the  District  Court 
of  the  United  States  for  the  Southern  District  of  New- York. 


G.iMlOPKlNS  &  SUM,  Pnnters. 


4 

TO 


CLEMENT    C.   MOORE,    LL.  D., 

SENIOR    TRUSTEE    OF     COLUMBIA    COLLEGE, 
THIS     WORK     IS     RESPECTFULLY      INSCRIBED, 
AS    A    TOKEN    OF    GRATITUDE    FOR    MANY 
AND    SIGNAL     FAVOURS, 

V 


*  ANI) 


OF    HIGH    RESPECT 


FOR    HIS    WORTH    AND    VIRTUES 


Jj  : 


PREFACE. 


THE  work  which  is  now  submitted  to  the  public,  comprises  a/portion 
of  the  materials  collected  for  the  courses,  it  is  my  duty  annually  to  deliver 
in  Columbia  College.  It  was  originally  intended  that  the  subject  of 
Practical  Mechanics,  should -have  accompanied  the  Elements  ;  thus 
forming  a  full  treatise  on  the  theoretic  and  practical  parts  of  that  useful 
and  interesting  branch  of  Science.  It  was  however  found  that  in  this 
way  the  work  would  have  assumed  too  bulky  a  form/  The  appli- 
cations of  the  elementary  principles  of  the  present  work  to  the  construc- 
tion of  Machines,  have  therefore  been  withheld,  untiVme  sense  of  the 
public  be  declared  as  to  its  merits.  Should  the  verdiet  be  favourable, 
the  author  may  be  encouraged  to  proceed  with  the  publication,  not  only 
of  the  sequel  to  these  elements,  but  with  that  of  som£  of  the  other  sub- 
jects to  which  a  labour  of  twelve  years  have  been  devyted,  namely :  Pure 
Physics,  Chemistry  applied  to  the  Arts,  and  Practical  Astronomy. 

In  the  discussion  of  subjects  so  extensive  and  Various,  the  author  is 
aware,  that  he  has  been  denied  the  advantages  tharare  to  be  derived  from 
the  division  of  labour,  and  has  been  unable  to  demote  to  any  one  object, 
that  steady  attention,  that  can  alone  ensure  entiri  success.  In  spite  of 
these  disadvantages,  he  ventures  to  submit  the/present  work  to  the  pub- 
lic, in  the  belief  that  it  cannot  fail  to  be  useful  to  the  student  of  Mechani- 
cal Science.  To  those  who  have  already  nude  progress,  this  volume 
may  present  little  novelty ;  all  that  it  contain/  of  most  value  will  be  rea- 
dily traced  to  obvious,  if  not  familiar  sources.)  But  to  the  learner,  he  can- 
not but  hope  that  it  will  offer  in  a  condensed,  md  generally  in  a  simple  and 
almost  popular  form,  facts,  principles,  and  Methods  of  investigation,  that 
he  will  find  in  no  single  work  in  any  language,  and  which  must  be  sought 
for  in  various  treatises,  most  .of  them  inaccessible  to  those  who  read  no 
ot  her  language  but  the  English. 


V1U  PREFACE.  ^    , 

The  work  presents  the  mixture  of  strict  mechanical  principles,  with 
the  physical  inferences  from  experiment  and  observation,  that  is  de- 
manded by  the  plan  of  teaching  Natural  Philosophy,  which  is  generally 
adopted  in  this  country,  and  which  is  habitual  in  most  of  the  English 
Treatises  on  that  general  subject.  It  is  therefore  a  combination  of  the 
subject,  styled  by  the  French  JVLecanique,  with  so  much  of  the  department 
called  by  them  Physique,  as  is  necessary  for  its  illustration,  and  for  pre- 
paring the  way  for  > practical  applications.  Nor  have  the  latter  been  wholly 
omitted.  They  have,  in  the  scope  of  the  treatise,  frequently  come  into 
view,  and  have  in  all  such  cases  been  treated  of  in  a  concise  manner. 

In  the  use  of  the  term  "  Mechanics",  it  has  been  employed  as  includ- 
ing the  whole  science  of  Equilibrium  and  Motion,  and  therefore  as  com- 
prising the  departments  of  Hydrostaticks  and  Hydrodynamicks. 

One  object  has  been  kept  steadily  in  view,  namely,  that  the  student 
shall  not  be  compelled,  after  having  mastered  this  treatise,  to  renew  his 
elementary  studies,  in  case  he  should  wish  to  rise  to  the  higher  applica- 
tions of  mechanical  science.  Should  the  author  not  have  failed  in  this, 
the  present  work,  however  humble,  in  the  limitation  of  its  direct  applica- 
tions to  the  mere  works  of  human  art,  may  serve  as  an  introduction  to  the 
science  that  investigates  the  mechanism  of  the  universe. 

In  order  to  obviaU  the  necessity  of  continually  quoting  the  authorities 
for  the  various  facts  md  investigations  employed  in  the  work,  by  which 
the  text  would  have  encumbered,  or  the  margin  loaded  with  notes,  a  list 
of  the  works  that  have  been  most  frequently  made  use  of  is  subjoined.  In 
some  few  cases,  where  foe  investigations  are  copied  without  alteration,  or 
where  they  are  not  compUte,  the  passage  of  the  author  quoted,  is  expressly 
referred  to  in  a  note.  This  list  of  authorities  is  far  from  complete ;  to  have 
made  it  so  would  have  appeared  rather  as  a  parade  of  research,  than  as  a 
security  from  the  charge  of  quoting  without  proper  acknowledgment ;  it  has 
therefore  been  principally  confined  to  those  writers  whose  labours  have 
not  become  in  some  measure  the  common  property  of  all  who  follow 
them  in  this  department  of  science. 

COLOMBIA  COLLEGE,  New- York.  February    Is7,  1832. 


LIST 

Of  the  Authors  principally  consulted  in  the  compilation  of  this  work. 

BARLOW,      .     .     .  *  .  On  Timber. 

Du  BUAT,     ....  Hydrodynamique. 

BIOT,       ..,.-.  Physique. 

BIOT  and  ARAGO,  .     .  Observations  Geodesiques. 

BOSSUT,        ....  Hydrodynamique. 

BOWDITCH,        .     .     .  Translation  of  Laplace. 

COULOMB,    ....  Machines  Simples. 

DANIELL,      ....  Atmospheric  Phenomena. 

DELAMBRE,       .     .     .  Base  du  Systeme  Metrique. 

GAUTHEY,    ....  Construction  des  Ponts. 

Do Canaux  de  Navigation. 

GENIEYS,      ....  Distribution  des  Eaux. 

HUTTON,      ....  Tracts  on  Gunnery. 

KATER, Report  on  Weights  and  Measures. 

LAPLACE,     ....  Mecanique  Celeste. 

NAVIER,       ....  Ponts  Suspendus. 

PECLET,        ....  Traite  de  Clialeur. 

PLAYFAIR,    ....  Outlines  of  Natural  Philosophy. 

POISSON,       .     .     .     .'  Mecanique. 

PONTECOULANT,     .     .  Systeme  du  Monde. 

POUILLET,    ....  Physique. 

PRONY,    .     .     .     .     .  Jaugage  des  Eaux. 

RICHARD,     ....  Manuel    d' Applications    Mathema- 

tiques. 

SABINE, Experiments  on  the  Pendulum. 

VENTURI,     ....  Motion  of  Fluids. 

VENTUROLI,       .     .     .  Mecanica. 

YOUNG  (Thomas),       .  Natural  Philosophy. 


TABLE  OF  CONTENTS. 

BOOK  I. 

OF  EQUILIBRIUM. 

CHAP.  I. 

General  Principles.  PAGE  1 

§1.  Definition  of  Mechanics. 

2.  Mode  of  determining  when  bodies  are  at  rest  or  in  mo- 

tion. 

3.  Co-ordinates  and  axes  to. which  the  position  of  points  is 

referred. 

4.  Change  in  the  length  and  position  of  the  co-ordinates  in 

case  of  motion. 

5.  Definition  of  Force. 

6.  Mode  of  measuring  forces. 

7.  Determination  of  the  direction  of  a  force. 

8.  Definition  of  Equilibrium. 

CHAP.    II. 

Equilibrium  of  Forces  acting  in  the  same  line.  4 

§    9.  Conditions  of  equilibrium  of  forces  acting  in  the  same 
line. 

10.  Definition  of  the  Resultant  of  Forces  and  its  Compo- 

nents: 

11.  Cases  of  the  relations  between  the  resultant  and  its 

components. 

CHAP.    III. 

Equilibrium  of  Forces  converging  to  a  point.    Composition  and 

Resolution  of  Forces.  6 

§  12.  Determination  of  the  resultant  of  two  forces  converg- 
ing to  a  point. 

13.  Composition  and  resolution  of  forces. 

14.  Conditions  of  equilibrium  among  three  forces. 

15.  Polygon  of  forces. 

16.  Determination  of  the  resultant  of  three  rectangular  for- 

ces converging  to  a  point. 


;i  TABLE  OF  CONTENTS. 

17.  Resolution  of  any  number  offerees  into  three  rectangular 

forces. 

18.  Conditions  of  equilibrium  in  a  system  of  forces,  each 

resolved  into  three,  at  right  angles  to  each  other. 

19.  Conditions  of  equilibrium  of  a  system  of  forces  acting 

upon  a  surface. 

CHAP.    IV. 

Equilibrium  of  Parallel  Forces.  PAGE  16 

§  20.  Method  of  proceeding  in  the  consideration  of  the  action 
of  parallel  forces. 

21.  Case  in  which  the  point  of  application  of  a  force  may 

be  conceived  to  be  changed. 

22.  Value  and  position  of  the  resultant  of  two  parallel 

forces. 

23.  Conditions  of  equilibrium  in  a  system  of  parallel  forces. 

24.  Definition  of  the  Centre  of  Parallel  Forces. 

25.  Investigation  of  formulae  for  finding  the  centre  of  pa- 

rallel forces. 

26.  Applications  of  the  foregoing  formula?. 

27.  Effect  of  the  connexion  of  points  into  rigid  systems. 

28.  Conditions  of  equilibrium  of  a  system  of  parallel  forces 

acting  upon  a  system  of  inflexible  and  inextensible 
lines. 

29.  Conditions  of  equilibrium  of  the  Funicular  Polygon  and 

Catenaria. 

CHAP.  v.  §- 

Equilibrium  of  Forces  in  the  same  plane,  but  neither  parallel,  nor 

converging  to  a  single  point.  33 

§  30.  General  determination  of  the  resultant  of  forces  acting 

in  the  same  plane. 
31  and  32.  Examination  and  definition  of  the  Moment  of 

Rotation. 

33.  Investigation  of  the  value  of  the  moment  of  rotation  of 
the  resultant  of  two  forces  in  terms  of  the  moments 
of  rotation  of  its  components. 

34  and  35.  Conditions  of  equilibrium  of  a  rigid  system 
having  one  fixed  point. 

36.  Extension  of  the  subject  to  forces  not  situated  in  one 

plane. 

37.  Definition  and  determination  of  the  value  of  the  projec- 

tion of  a  force. 

38.  Extension  of  the  conditions  of  equilibrium  of  a  rigid  sys- 

tem to  points  connected  in  any  manner  whatsoever. 


TABLE  OF  CONTENTS.  XI11 

BOOK  II. 

OF  MOTION^ 

CHAP.   I. 

Of  motion  in  general.     Uniform  motion.     General  principles 

of  variable  motion.  PAGE  39 

§  39.  Principle  of  Inertia. 

40.  Definition  of  Velocity ;  its  value  in  terras  of  the  space 

and  time. 

41.  Comparison  of  uniform  motions  in  the  same  line. 

42.  Composition  and  resolution  of  motion. 

43.  Variable  motions  and  the  forces  that  cause  them. 

44.  Accelerating  forces. 

45.  Relation  between  the  force  and  velocity  in  uniform  mo- 

tion. 

46.  Investigation  of  the  circumstances  of  variable  motion. 

CHAP.    II. 

Of  Rectilineal  motion  uniformly  accelerated,  or  uniformly  re- 
tarded. 46 
§  47.  Definition  of  uniformly  accelerated  or  retarded  motion. 

48.  Definition  of  final  velocity. 

49.  Investigation  of  the  circumstances  of  uniformly  accele- 

rated motion. 

50.  Investigation  of  the  circumstances  of  uniformly  retarded 

motion. 

51.  Relations  of  the  forces  that  cause  curvilinear  motion.    N 

CHAP.    III. 

Of  Curvilinear  Motion.  50 

§  52.  Investigation  of  the  general  circumstances  of  curvilinear 
motion. 

CHAP.    IV. 

Of  Parabolic  Motion.  52 

§53.  Investigation  of  the  circumstances  of  the  motion  of  a 
body  acted  upon  by  two  forces  inclined  to  each  other ; 
of  which  one  would  produce  uniform  motion,  and 
the  other  is  a  constant  force,  always  acting  parallel 
to  itself. 

CHAP.    V. 

Of  the  Motion  of  Points  compelled  to  move  on  surfaces  by  the 

action  of  accelerating  forces.  56 

§  54.  Method  of  considering  the  action  of  a  surface  on  bodies 
compelled  to  move  upon  it. 


XJV  TABLE  OF  CONTENTS. 

55.  Case  of  motion  on  a  surface  inclined  at  a  constant 

angle  to  the  direction  of  an  accelerating  force,  always 
acting  parallel  to  itself. 

56.  Investigation  of  the  circumstances  of  motion  on  a  plane 

surface  under  the  action  of  an  accelerating  force 
always  parallel  to  itself. 

57.  Circumstances  of  motion  on  a  system  of  planes. 

58.  Circumstances  of  motion  on  a  curve. 

59.  Circumstances  of  motion  on  a  cycloid. 

60.  Circumstances  of  motion  in  a  circular  arc. 

61.  Definitions  of  Central  Forces. 

62.  Cause  of  motion  in  circular  orbits. 

63.  Uniformity  of  the  velocity  in  circular  orbits. 

64.  Circumstances  of  motion  in  circular  orbits. 

65.  Relation  between  the  periodic  times  and  distances  in 

circular  orbits. 

66.  General  principles  of  motion  in  curvilinear  orbits. 
67  and  68.  Principle  of  areas. 

CHAP.    VI. 

Principle  of  D'Alembert.  PAGE  71 

§  69.  Enunciation  and  illustration  of  the  principle  of  D'  Alem- 
bert. 

CHAP.    VII. 

Principle  of  Virtual  Velocities.  72 

§  70.  Definition  of  Virtual  Velocities. 
71.  Illustrations  of  the  principle  of  virtual  velocities. 


BOOK  III. 

OF  THE  EQUILIBRIUM  OF  SOLID  BODIES. 
CHAP.    I. 

General  properties  of  Matter.     Division  of  Natural  Bodies. 

Measure  of  the  Moving  Force  oj  Bodies.  77 

§  72.  Essential  properties  of  matter. 

73.  Extension  of  matter. 

74.  Divisibility  of  matter. 

75.  Definition  of  Body.     Indestructibility  of  matter. 

76.  Atomic  theory  stated. 

77.  Impenetrability  of  matter. 

78.  Mobility  of  matter ;    relations  of  the  Density,  Volume, 

and  Mass  of  bodies. 

79  and  80.  Division  of  bodies  according  to  their  mechani- 
cal states. 


TABLE  OF  CONTENTS.  XV 

81.  Forces  that  determine  the  mechanical  state  of  bodies. 

82.  Division  of  the  subject  growing  out  of  the  different  me- 

chanical states  of  bodies. 

83.  Quantity  of  motion  in  bodies. 

CHAP.    II. 

Attraction  of  Gravitation.  PAGE  83 

§  84.  Fall  of  heavy  bodies  near  the  earth. 

85.  Direction  of  the  force  of  gravity. 

86.  Convergence  of  these  directions. 

87.  Relation  of  the  attraction  of  gravitation  to  the  quantity  of 

matter  and  velocity. 

88.  Question  in  relation,  to  the  mutual  attraction  of  bodies. 

89.  Experiment  of  Schehallien. 

90.  Principles  of  the  experiment  of  Cavendish. 

9 1 .  Description  of  the  apparatus  of  Cavendish. 

92.  Principles  on  which  the  mass  of  the  planets  is  deter- 

mined. 

93.  Consideration  of  gravity  as  an  accelerating  force. 

94.  Experiments  of  Galileo.  , 

95.  Experiments  and  machine  of  Atwood. 

96  and  97.  Proofs  of  the  mutual  action  of  the  earth  and 
moon.  Law  according  to  which  the  intensity  of  gra- 
vity decreases. 

98.  Proofs  of  the  universal  influence  of  gravitation* 

99.  Force  of  gravity  cannot  be  considered  constant,  even 

at  small  distances  from  the  surface  of  the  earth. 

100.  Effect  of  the  earth's  rotation  on  the  force  of  gravity  at 

the  earth's  surface. 

101.  Investigation  of  the  relation  between  the   centrifugal 

force  at  the  equator,  and  the  whole  force  of  gravity. 

102.  Effect  of  the  combined  action  of  gravitation  and  the  cen- 

trifugal force,  on  the  figure  of  the  earth. 

103.  Recapitulation  of  the  laws  of  universal  gravitation. 

CHAP.  III. 

Of  the  Centres  of  Gravity  and  Inertia.  100 

§  104.  Direction  of  the  force  of  gravity.    Definition  of  Weight. 

105.  Definition  of  the  Centre  of  Gravity.     Application  of  the 

conditions  of  equilibrium  of  parallel  forces,  to  the 
determination  of  the  position  of  the  centre  of  gravity. 

106.  Conditions  of  equilibrium  of  a  body  supported  by  a  point. 

107.  Conditions  of  equilibrium  of  a  body  resting  on  an  edge. 
108  and  109.   Conditions  of  equilibrium  of  a  body  resting  on 

a  surface. 

110.  Effect  of  the  figure  of  bodies  on  their  stability. 

111.  Stability  of  bodies  resting  on  inclined  surfaces. 

11?.  Conditions  of  equilibrium  in  bodies  resting  on  points. 


XVI  TABLE  OF  CONTENTS, 

113.  Phenomena  of  rolling  and  sliding  bodies. 

114.  Experimental  methods  of  determining  the  position  of 

the  centre  of  gravity. 

115.  Definition  of  Centre  of  Inertia. 

••  «. 

CHAP.    IV. 

Of  Friction.  PAGE  112 

§  116.  Resistances  to  the  motion  of  bodies. 

117.  Reference  to  the  authors  who  treat  of  friction. 

118.  Varieties  of  friction. 

119.  Methods  of  determining  the  friction  that  prevents  bodies 

being  set  in  motion. 

120.  Experiments  and  inferences  in  relation  to  the  friction  of 

moving  bodies. 

121.  Friction  of  rolling  bodies. 

122.  Friction  at  the  axles  of  wheels. 

123.  Theory  of  Coulomb. 

124.  Formula  that  expresses  the  results  of  the  experiments  on 

friction. 

125.  Methods  of  lessening  and  overcoming  friction. 

126.  Effects  of  friction  in  bringing  bodies,  moving  near  the 

surface  of  the  earth,  to  rest. 

CHAP.  v. 

Of  the  Stiffness  of  Ropes.  124 

j§  127.  Circumstances  on  which  the  stiffness  of  ropes  depends. 

128.  Results  of  the  experiments  of  Coulomb. 

129.  Effect  of  increasing  the  number  of  turns  made  by  a  rope 

around  a  cylinder. 

130.  Data  for  calculating  the  stiffness  of  ropes  obtained  from 

the  experiments  of  Coulomb. 

CHAP.    VI. 

Of  the  Mechanic  Powers.  126 

§  131.  Definition  of  Machines.      General  principle   of  their 
action.    * 

132.  Definition  of  the  mechanic  powers. 

133.  Number  and  classification  of  the  mechanic  powers. 

Of  the  Lever.  127 

§  134.  Definition  of  the  Lever. 

135.  General  condition  of  equilibrium  in  the  lever. 

136.  Definition  of  the  Power  and  the  Weight.    Classification 

of  the  different  kinds  of  lever. 

137.  Condition  of  equilibrium  in  a  straight  lever,  on  which 

the  power  and  weight  act  parallel  to  each  other. 

138.  Condition  of  equilibrium  of  a  lever,  whether  straight  or 

crooked  in  any  direction  of  the  power  and  weight. 


TABLE  OF  CONTENTS.  XV11 

139.  Examples  of  the  use  of  the  different  kinds  of  lever. 

140.  Principles  on  which  the  properties  of  the  balance  depend. 

141.  Error  arising  from  inequality  of  the  arms  of  a  balance. 

Mode  of  counteracting  and  correcting  that  error. 

142.  Properties  of  a  good  balance. 

143.  Condition  of  equilibrium  in  a  compound  system  of  levers. 

144.  Description  and  theory  of  the  platform  balance. 

145.  Description  and  theory  of  the  steelyard. 

146.  Investigation  of  the  effect  of  friction  upon  the  lever. 

Of  the  Wheel  and  Axle.  PAGE  138 

§  147.  Condition  of  equilibrium  in  the  Wheel  and  Axle.  Various 
forms  of  the  wheel  and  axle. 

148.  Combinations  of  wheels  and  axles,  by  means  of  bands. 

149.  Wheels  and  Pinions. 

1 50.  Examination  of  the  proper  form  to  be  given  to  the  teeth  of 

wheels  and  pinions. 

1 51.  Change  of  intensity  of  force,  and  of  velocity  produced  in 

the  action  of  wheels  upon  pinions,  and  of  pinions  upon 
wheels. 

152.  Change  in  the  direction  of  motion,  produced  by  different 

forms  of  the  wheel  and  pinion. 

153.  Effect  of  friction  upon  the  equilibrium  of  the  wheel  and 

axle. 

154.  Allowance  for  friction  in  the  action  of  wheels  and  pin- 

ions. 

Of  the  Pulley.  146 

§  155.  Conditions  of  equilibrium  in  the  fixed  and  moveable 
Pulley. 

156.  Conditions  of  equilibrium  in  combinations  of  pullies. 

157.  Effect  of  a  want  of  parallelism  in  the  ropes  on  the  equi- 

librium of  pullies. 

158.  Friction  of  pullies,  and  modes  of  lessening  it. 

Of  the  Wedge.  151 

159.  Definition  of  the  Wedge. 

160.  Condition  of  equilibrium  in  the  isosceles  wedge. 

161.  Practical  action  of  the  wedge. 

162.  Valuable  applications  of  the  wedge. 

163.  Extension  of  the  principle  of  the  wedge  to  bodies  of 
'•-&         other  figures. 

164  and  165.  Effect  of  friction  on  the  action  of  the  wedge. 

Of  the  Inclined  Plane.  1 55 

§  166.  Definition  of  the  Inclined  Plane,  and  mode  in  which 
the  forces  act  upon  it. 

167.  Conditions  of  equilibrium  in  the  inclined  plane. 

168.  Applications  of  the  inclined  plane. 

169.  Effect  of  friction  on  the  equilibrium  of  the  inclined  plane, 

3* 


TABLE  OF  CONTENTS. 

Of  the  Screw.  PAGE  157 

§  170.  Mode  of  forming  the  Screw,  and  conditions  of  its  equi- 
librium. 

171.  Effects  of  friction  on  the  action  of  the  screw. 

172.  Endless  screw. 

173.  Valuable  applications  of  the  screw. 

174.  Hunter's  improvement  hi  the  screw. 

175.  Effect  of  friction  on  the  endless  screw,  when  combined 

with  the  wheel  and  axle. 

CHAP.    VII. 

Of  the  Strength  of  Materials.  165 

§  176.  Method  of  proceeding  in  considering  the  strength  of 
materials. 

177.  Hypothesis  of  Galileo. 

178.  Different  modes  in  which  forces  act  to  destroy  the  aggre- 

gation of  bodies. 

Of  the  Absolute  Strength  of  Materials.  166 

§  179.  Analytic  expressions  for  the  absolute  strength  of  mate- 
rials. 

180.  Inferences  from  experiments,  and  tables  of  the  absolute 
strength  of  different  substances. 

Of  the  Respective  Strength  of  Materials.  169 

§  181.  Inferences  from  the  hypothesis  of  Galileo,  in  respect  to 
beams  supported  at  one  end. 

182.  Inferences  from  the  same  hypothesis  in  respect  to  beams 

resting  on  two  props. 

183.  Inferences  in  respect  to  beams  firmly  fixed  to  each  end. 

184.  Examination  of  the  effect  of  the  inclination  of  a  beam  to 

the  horizon. 

185.  Consequences  of  an  uniform  distribution  of  the  weight 

that  acts  upon  a  beam. 

186.  Action  of  the  beam's  own  weight. 

187.  Comparison  between  the  inferences  from  the  hypothesis 

and  the  results  of  experiment,  in  the  cases  where  they 
nearly  agree. 

188.  Discrepancies  between  the  inferences  from  the  hypothe- 

sis, and  the  results  of  experiment. 

189.  Tables  of  the  respective  strength  of  various  substances. 

Of  the  Resistance  of  Bodies  to  a  Force  exerted  to  crush  them.       181 

§  190.  Expressions  for  the  resistance  of  a  rectangular  pillar  to  a 

crushing  force. 

191.  Tables  of  the  resisting  force  of  various  bodies  to  a 
crushing  force. 


TABLE  OF  CONTENTS.  XIX 

Of  the  Strength  of  Torsion.  PAGE  183 

§  1 92.  Laws  which  the  resistance  to  Torsion  follows. 
1 93.  Relative  resistances  to  torsion  of  different  substances. 

CHAP.    VIII. 

Of  the  Equilibrium  of  Artificial  Structures.  184 

§  194.  Principle  on  which  the  equilibrium  of  artificial  struc- 
tures is  investigated. 
195.  Definition  of  the  Strength,  and  of  the  Stress  or  Thrust. 

Of  the  Equilibrium  of  Walls.  184 

§  196.  Conditions  of  equilibrium  of  a  prismatic  wall. 

197.  Effect  of  friction  on  the  conditions  of  equilibrium. 

198.  Conditions  of  equilibrium  of  walls  with  sloping  faces, 

and  with  buttresses. 

199.  Applications  of  the  theoretic  principles. 

Equilibrium  of  Columns.  187 

§  200.  Investigation  of  the  conditions  of  equilibrium  in  columns. 

201.  Inferences  from  the  investigation  of  §  200.      Figure  of 

columns  loaded  with  weights. 

202.  Effect  of  a  lateral  thrust  in  the  weight  that  acts  on  a  col- 

umn. 

203.  Figure  of  a  column  intended   to   resist  the  action  of  a 

fluid. 

Equilibrium  of  Terraces.  196 

§  204.  Mode  in  which  earth  acts  on  a  terrace  wall. 
205.  Investigation  of  the  conditions  of  equilibrium. 

Equilibrium  oj  Arches.  198 

§206.   Reasons  for  the  use  of  Arches.     Different  materials  of 
which  they  may  be  constructed. 

207.  Definitions  and  names  of  the  parts  of  an  arch. 

208.  Former  mode  of  proceeding  in  the  investigation  of  the 

theory  of  arches.     Its  defects. 

209.  Manner  in  which  arches  are  affected  by  the  mutual 

action  of  their  parts. 

210.  Investigation  of  the  position  of  the  points  of  rupture  in 

an  arch. 

211.  Inferences  from  the  investigation  in  §210. 

212.  Relative  thicknesses  of  piers  and  abutments. 

213.  List  of  the  most  remarkable  stone  arches. 

Equilibrium  of  Dom  es.  207 

§  214.  Comparison  between  domes  and  cylindric  vaults. 

215.  Geometric  properties  of  domes. 

216.  Description  of  some  remarkable  domes. 


XX  TABLE  OF  CONTENTS. 

Of  Wooden  Arches.  PAGE  210 

§  217.  Comparison  between  the  actions  of  wood  and  stone, 
when  employed  in  spanning  openings. 

218.  Strength  of  trussed  beams. 

219.  Strength  of  a  combination  of  two  straight  beams. 

220.  Strength  of  a  wooden  arc  of  small  curvature. 

221.  Strength  of  a  wooden  arc  of  greater  curvature. 

222.  Applications  to  the  construction  of  bridges. 

223.  Principle  applied  by  Grubenman,  in  the  construction  of 

wooden  bridges. 

Of  Cast  Iron  Arches.  218 

§  224.  Comparison  between  the  actions  of  cast  iron,  and  those 

of  wood  and  stone  in  the  construction  of  arches. 
225.  Instances  of  cast  iron  arches. 

Of  Chain  Bridges.  219 

§  226.  Progress  of  the  art  of  erecting  chain  bridges. 
227.  Theory  of  chain  bridges. 
228    to  234.  Inferences  made  by  Navier,  from  the  theory. 

238.  Different  cases  in  which  the  principle  may  be  applied. 

239.  Notice  of  wire  bridges. 


BOOK  IV. 
OF  THE  MOTION  OF  SOLID  BODIES. 

CHAP.    I. 

Of  Falling  Bodies.  227 

§  237.  Action  offerees  whose  direction  passes  through  the  cen- 
tre of  gravity  of  a  solid  body. 

238.  Investigation  of  the  circumstances  of  the  motion  of  a  body 

falling  from  rest  near  the  surface  of  the  earth,  the 
resistance  of  the  air  being  left  out  of  view. 

239.  Circumstances  of  the  motion  of  a  body  projected  verti- 

cally upwards. 

240.  Investigation  of  the  effect  produced  by  the  resistance  of 

the  air,  on  the  hypothesis  that  it  varies  with  the  square 
of  the  velocity. 

241.  Discrepancies  between  the  hypothesis  used  in  §  240,  and 

the  actual  law  of  the  resistance  of  the  air. 

CHAP.  II. 

Of  the  Rotary  Motion  of  Bodies.  232 

§  242.  Theory  of  the  Moment  of  Inertia. 


TABLE  OP  CONTENTS.  XXI 


243.  Properties  of  the  Centre  of  Gyration,  and  modes  of 

finding  it. 

244.  Properties  of  the  Centre  of  Percussion. 

245.  Consideration  of  the  double  motion  of  translation  and 

rotation  produced  by  the  action  of  a  force  whose  direc- 
tion does  jiot  pass  through  the  centre  of  gravity  of  a 
solid  body. 

246.  Proposition  in  relation  to  the  three  principal  axes  of  rota- 

tion. 

247.  Variation  in  the  velocity  of  the  points  of  a  body  endued 

with  a  double  motion  of  rotation  and  translation,  when 
this  velocity  is  estimated  in  the  direction  in  which  the 
centre  of  gravity  moves. 

248.  Centre  of  spontaneous  rotation.     Effects  of  the  varying 

velocity,  spoken  of  in  §  247,  on  bodies  moving  in 
resisting  media. 

249.  Applications  of  the  theory  of  the  moment  of  inertia. 

250.  Mode  of  finding  the  moment  of  inertia  in  respect  to  a 

given  axis,  when  this  moment  is  given  in  respect  to 
another  line  considered  as  an  axis. 

251.  Applications  of  §250. 


§  252. 


CHAP.    III. 

Motion  of  Projectiles.  PAGE  242 

Circumstances  of  the  motion  of  projectiles,  abstracting 


the  resistance  of  the  air. 

253.  Investigation  of  the  effect  of  a  resisting  medium ;  appli- 

cation to  the  resistance  of  the  air,  upon  the  hypothesis 
that  it  varies  with  the  square  of  the  velocity. 

254.  Application  to  military  projectiles.     Mode  of  action  of 

inflamed  gunpowder.  Inferences  from  the  experi- 
ments of  Hutton,  in  respect  to  the  expansive  force  of 
gunpowder. 

255.  Effect  of  confined  gunpowder  according  to  the  experi- 

ments of  Rumford.  Consequences  that  follow  in 
certain  cases. 

256.  Citation  of  the  best  experiments  on  military  projectiles. 

257.  Original  mode  of  determining  the  initial  velocity  of  a  pro- 

jectile. 
"258.  Ballistic  pendulum  of  Robins. 

259.  Effect  of  the  length  of  the  piece  on  the  initial  velocity. 

260.  Relation  of  the  initial  velocities  to  the  charges  of  gun- 

powder, and  the  weight  of  the  balls. 

261.  Results  ^f  Hutton' s  experiments  on  the  resistance  of  the 

air.     Beduction  of  formulae  for  practical  purposes. 

262.  Applications  of  the  formulas  of  §  261. 

263.  Inferences  from  the  theory  in  respect  to  the  maximum 

charge  of  gunpowder. 

264.  Inferences  in  respect  to  the  windage. 


TABLE  OP  CONTENTS. 

265.  Deviation  of  projectiles  from  the  vertical  plane,  and  its 

cause. 

266.  Advantages  of  rifling  the  bores  of  guns. 

267.  Improvements  in  military  ordnance  resulting  from  the 

experiments  of  Hutton. 

268.  Inquiry  into  the  best  figure  for  cannon. 

269.  Penetration  of  military  projectiles  into  solid  bodies. 

Comparison  of  ancient  and  modern  military  engines. 

270.  Principles  of  the  Ricochet  and  its  applications. 

CHAP.  iv. 
Theory  of  the  Pendulum.  PAGE  267 

§  271.  Circumstances  of  the  motion  of  a  Pendulum. 

272.  Theory  of  the  motion  of  a  simple  pendulum  in  a  circular 

arc. 

273.  Isochronism  of  a  simple  pendulum  moving  in  cycloidal 

arcs.  Method  proposed  to  cause  pendulums  to  move 
in  a  cycloid. 

274.  Effect  of  difference  of  distance  from  the  surface  of  the 

earth  on  the  lengths  of  isochronous  pendulums. 

275.  Effect  of  the  variation  of  the  force  of  gravity  on  the 

lengths  of  isochronous  pendulums  in  different  lati- 
tudes. 

276.  Definition  of  the  compound  pendulum ;  properties  of  the 

Centre  of  Oscillation. 

277.  The  centres  of  suspension  and  oscillation  are  convertible 

points. 

278.  Investigation  of  the  position  of  the  centre  of  oscillation, 

in  homogeneous  bodies  of  certain  figures. 

279.  Effects  of  the  resistance  of  the  air  on  the  motions  of  pen- 

dulums. 

CHAP.  v. 

Applications  of  the  Pendulum.  277 

§  280.  Various  purposes  to  which  the  pendulum  is  applicable. 

281.  Principles  of  the  application  of  the  pendulum  as  a  mea- 

sure of  time. 

282.  Description  and  theory  of  compensation  pendulums. 

283.  Methods  in  use  for  measuring  the  length  of  the  pendu- 

lum. 

284.  Method  of  Borda. 

285.  Reduction  of  the  pendulum's  vibrations  to  a  cycloidaJ 

arc ;  corrections  for  the  air's  resistance ;  determina- 
tion of  the  length  of  the  seconds  pendulum  from  tint 
of  the  experimental ;  reduction  to  thp  level  of  th* 
sea  ;  effect  of  the  figure  and  density  n  the  grounJ  •"' 
which  the  experiment  is  made. 

286.  Method  of  Kater. 

287.  Deduction  of  the  measure  of  the  force  of  grav/ry  fro 

length  of  the  second's  pendulum. 


TABLE  OF  CONTENTS.  XX111 

288    and  289.    Comparison  of  the  methods  of  Borda  and 
Kater. 

290.  Results  of  the  measures  of  the  pendulum  in  different 

places. 

291.  Principle  on  which  the  pendulum  is  applied  as  a  stand- 

ard of  measure. 

292.  British  system  of  weights  and  measures. 

293.  System  of  the  state  of  New- York. 

294.  Comparison  of  the  two  foregoing  systems. 

295.  French  system  of  weights  and  measures. 

296.  Examination  of  the  advantages  and  defects  of  the  French 

system. 

297.  Application  of  the  principle  of  the  pendulum  in  Caven- 

dish's experiment. 

CHAP.    VI. 

Of  Collision.  PAGE  300 

§  298.  Division  of  bodies  into  elastic  and  non-elastic. 

299.  Investigation  of  the  consequences  of  the  collision  of  non- 

elastic  bodies  of  spherical  figure  moving  in  the  same 
straight  line. 

300.  Oblique  impact  of  a  non-elastic  body  against  an  immovea- 

ble  plane. 

301.  Oblique  impact  of  non-elastic  bodies  of  spherical  figures. 

302.  Impact  of  non-elastic  bodies  moving  in  parallel  lines. 

303.  Investigation  of  the  consequences  of  the  collision  of 

elastic  bodies  of  spherical  figure  moving  in  the  same 
straight  line. 

304.  Recapitulation  of  the  inferences  from  the  foregoing  in- 

vestigation. 

305.  Oblique  impact  of  an  elastic  body  against  an  immovea- 

ble  surface. 

306.  Constancy  of  the  sum  of  the  vires  viva  in  the  collision 

of  perfectly  elastic  bodies. 

307.  Gain  of  motion,  estimated  in  a  given  direction,  in  the  col- 

lision of  a  small  elastic  body  against  a  larger  one. 

308.  Effects  of  impact  when  the  elastic  bodies  do  not  move 

in  the  same  straight  line. 

309.  The  state  of  the  common  centre  of  gravity  in  respect  to 

motion  or  rest  is  not  affected  by  the  collision  of  bo- 
dies. 

310.  Investigation  of  the  effect  of  imperfect  elasticity. 

BOOK  V. 

OF  THE  EQUILIBRIUM  OF  FLUIDS. 
CHAP.    I. 

General  Characters  of  Fluid  Bodies.  309 

§  311.  Distinctive  property,  and  classification  of  fluids. 


XXIV  TABLE  OF  CONTENTS. 

312.  Characters  of  liquids. 

313    and  314.  Investigation  of  the  general  conditions  of  the 

equilibrium  of  fluids. 
315.  Difference  in  the  action  of  solid  and  fluid  masses. 

CHAP.    II. 

Equilibrium  of  Gravitating  Liquids.  PAGE  314 

§  316.  Condition  of  equilibrium  in  gravitating  liquids. 

317.  Figure  of  the  surface  of  gravitating  liquids. 

318.  Definition  of  the  art  of  Levelling. 

319.  Description  of  the  Water  Level. 

320.  Description  of  the  Spirit  Level. 

321.  Mode  of  using  the  spirit  level. 

322.  Description  of  the  Mason's  Level. 

323.  Correction  of  level  for  the  sphericity  of  the  earth. 

324.  Effect  of  atmospheric  refraction  on  levelling. 

325.  Equilibrium  of  a  homogeneous  liquid  in  a  bent  tube. 

326.  Equilibrium  of  heterogeneous  liquids. 

327.  Effects  produced  on  the  weight  of  solids  immersed  in 

fluids. 

328.  Equilibrium  of  a  s'olid  floating  at  the  surface  of  a  liquid. 

329.  Comparison  of  the  immersion  of  the  same  body  floating 

on  different  liquids. 

CHAP.  in. 

Of  the  Pressure  of  Gravitating  Liquids.  325 

§  330.  Investigation  of  the  expressions  for  the  pressure  of  a 
liquid  on  a  surface. 

331.  Inferences  from  the  foregoing  investigation. 

332.  Action  of  a  liquid  to  change  the  direction  and  intensity 

of  the  forces  that  act  upon  it. 

333.  Hydrostatic  paradox. 

334.  Comparison  of  the  theory  with  the  results  of  experiment. 

335.  Properties  of  the  centre  of  fluid  pressure. 

336.  Investigation  of  the  amount  of  pressures  exerted  by  a 

liquid  on  a  body  immersed  in  it. 

CHAP.    IV. 

Of  Specific  Gravity.  333 

§  337.  Definition  of  Specific  Gravity  ;  standard  in  which  it  is 
estimated. 

338.  Changes  produced  in  the  density  of  water  by  variations 

of  temperature. 

339.  Principles  of  the  method  of  specific  gravities. 

340.  Modes  of  determining  the  specific  gravity  of  solid  bodies. 

341.  Modes  of  determining  the  specific  gravity  of  liquids. 

342.  Method  of  ascertaining  the  weight  of  given  bulks  of 

liquids. 


TABLE  OF  CONTENTS.  XXV 

348.  Determination  of  the  specific  gravities  ofliquids  by  glass 

bubbles. 

349.  Table  of  Specific  Gravities. 

350.  Application  of  the  Method  of  specific  gravities  to  deter- 

mine the  volume  of  bodies. 

CHAP.  v. 

Of  the  nature  and  characters  of  Elastic  Fluids,  and  of  the 

Pressure  of  the  Jltmosphere.  PAGE  347 

351.  Modes  in  which  we  become  acquainted  with  the  exist- 

ence of  Elastic  Fluids. 

352.  Distinctive  properties  of  permanently  Elastic  Fluids  or 

Gases. 

353.  Characters  of  condensable  elastic  fluids  or  Vapours. 

354.  Effect  of  heat  on  the  tension  of  elastic  fluids. 

355.  Law  of  Dalton. 

356.  Experiment  of  Torricelli. 

357.  Estimate  of  the  amount  of  atmospheric  pressure. 

358.  Principles  of  action,  and  description  of  the  common  pump 

and  syphon. 

CHAP.    VI. 

Of  the  Air  Pump.  359 

359.  Description  of  the  Air  Pump. 

360.  Gauges  for  the  air  pump. 

361.  Proofs  afforded  by  the  air  pump  of  the  pressure  of  the 

atmosphere.  / 

362.  Experimental  proof  that  the  pressure  *<  the  air  causes 

the  rise  of  liquids  in  the  pump  and  in  the  Torricellian 
apparatus. 

363.  Experimental  proof  that  the  air  has  weight. 

364.  Experimental  proofs  of  the  elasticity  of  air. 

CHAP.  vn. 
Equilibrium  of  Permanently  Elastic  Fluids.  360 

365.  Experimental  investigation  of  the  law  of  Mariotte. 

366.  Conditions  of  the  equilibrium  of  a  gaseous  atmosphere  of 

uniform  temperature. 

367.  Law  of  Gay-Lussac. 

368.  Investigation  of  the  relation  between  the  densities  and 

volumes  of  gases  under  different  circumstances  of 
temperature  and  pressure. 

369.  Properties  common  to  liquids  and  elastic  fluids ;  buoy- 

ancy of  the  atmosphere. 

4* 


XX Vi  TABLE  Of  CONTENTS. 

CHAP.    VIII. 

Of  the  Equilibrium  and  Tension  of  Vapours.       PAGE  368 
§  370.  Vapour  is  formed  at  other  temperatures  than  that  at  which 
ebullition  occurs. 

371.  Relation  of  the  weight  and  tension  of  vapour  in  a  given 

space,  to  the  temperature  and  pressure. 

372.  Effect  of  unequal  temperature,  in  the  space  occupied  by 

vapour,  on  its  tension. 

373.  Expression  for  the  relations  between  the  temperatures, 

tensions,  and  densities  of  vapours. 

374.  Experimental  determinations  of  the  tension  of  aqueous 

vapour  at  different  temperatures. 

375.  Dalton's  Law  of  the  tension  of  vapours  of  different  sub- 

stances. 

376.  Tension  of  mixtures  of  gases  and  vapours. 

CHAP.    IX. 

Of  the  Specific  Gravity  of  Elastic  Fluids.  374 

§  377.  Method  of  determining  the  specific  gravity  of  gases. 
Table  of  the  specific  gravity  of  gases. 

378.  Method  of  determining  the  specific  gravity  of  vapours. 

Table  of  the  specific  gravity  of  vapours. 

379.  Table  of  the  density  and  volume  of  steam  at  different 

temperatures. 

380.  Of  the  Dew-Point,  and  method  of  finding  it. 

CHAP,    X. 

Of  the  Barometer  and  its  Applications.  379 

§  381.  Various  forms  and  modifications  of  the  Barometer. 

382.  Theory  of  the  Vernier. 

383.  Modes  of  rendering  the  barometer  portable. 

384.  Method  of  filling  and  graduating  barometers. 

385.  Accidental  and  periodic  variations  in  atmospheric  pres- 

sure, and  the  height  of  the  barometer. 

386.  Influence  of  the  accidental  variations  on  the  weather. 

387.  Principle  on  which  the  barometer    is  applied  in  the 

mensuration  of  heights.  Effect  of  variations  of 
temperature  on  the  results,  and  correction  for  differ- 
ence of  temperature. 

388.  Rules  for  observation,  and  for  calculating  heights  by  the 

barometer. 

389.  Correction  for  moisture,  and  for  the  latitude  of  the  place. 

Example  of  the  form  of  calculation. 


TABLE  OP  CONTENTS,  XXVH 

CHAP.  XI. 

Of  the  Attraction  of  Cohesion.  PAGE  390 

§  390.  Experiment  showing  the  existence  of  the  Attraction  of 

Cohesion.  % 

391.  Phenomena  arising  from  the  attraction  of  cohesion. 

392.  Theory  of  Capillary  Attraction. 

393.  Relation  between  the  forces  of  cohesion  and  aggrega- 

tion. 

394.  Relation  between  the  heights  to  which  a  liquid  rises  in 

tubes  of  different  diameters,  in  concentric  tubes,  and 
between  plates. 

395.  Effect  of  capillary  action  on  aeriform  fluids. 


BOOK  VI. 

OF  THE  MOTION  OF  FLUIDS.       . 
CHAP.    I. 

Theory  of  the  Motion  of  Liquids.  396 

§  396.  Hypothesis  used  in  examining  the  motion  of  liquids. 
397.  Examination  of  the  manner  in  which 'liquids  actually 
move. 

CHAP.    II. 

Of  the  Motion  of  Liquids  through  orifices  pierced  in  thin 

plates.  400 

§  398.  Theory  of  this  motion. 

399.  Shape  of  the  jet  determined  by  experiment.     InAv-nce 

of  the  contraction  of  the  vein  on  the  theoretic-esu^g' 

400.  Phenomena  of  spouting  liquids. 

401.  Formulae  for  the  height  and  amplitude  of  a  i*<  of  liquid. 

402.  Theory  of  the  vortex  formed  in  the  disch^'ge  of  bquids 

from  vessels. 

CHAP.    III. 

Of  the  Discharge  of  Liquids  thr^gh  short  r*pes  or  adjut- 
age- 

§  403.   General  effect  produced  >•/  applying  *  pipe  to  an  orifice 
through  which  a  lia-ld  issues. 

404.  Effects  of  adjutage*  of  differed  forms. 

405.  Table  of  the  ejects  of  adjudges. 


XXVlll  TABLE  OP  CONTENTS. 

CHAP.    IV. 

Of  the  Motion  of  Water  in  Pipes.  PAGE  4  1  1 

§  406.  Theory  of  the  motion  of  water  in  pipes  of  uniform  bore. 

407.  Effect  of  variations  in  the  diameter  of  the  pipes. 

408.  Obstructions  to  which  pipes  are  liable,  and  modes  of  ob- 

viating them. 

409.  Effect  of  bends  or  elbows  in  the  pipes. 

410.  Theory  of  the  motion  of  water  -in  pipes  diverging  from  a 

main-pipe. 

CHAP.  v. 
Of  the  Motion  of  Liquids  in  Open  Channels.  421 

§  411.  Theory  of  the  motion  of  liquids  in  open  channels  of  uni- 
form section. 

412.  Relations  between  the  velocity  at  the  surface,  and  the 

mean  velocity  of  a  stream. 

413.  Effect  of  variations  in  the  section  of  a  stream,  and  of 

bends  in  its  course.    Principle  of  the  lateral  commu- 
nication of  motion  in  fluids. 

CHAP.    VI. 

Of  Rivers.  42J 

§  414.  Circulation   of  aqueous  matter  between  the  land  and 
ocean. 

415.  Relation  between  the  beds  of  rivers,  and  the  force  of 

their  currents. 

416.  Effects  of  the  overflow  of  rivers. 

417.  Obstructions  to  which  the  navigation  of  rivers  is  liable. 
x  4  IS.  Modes  of  removing  such  obstructions. 

419\Different  methods  of  gauging  streams. 

CHAP.     VII. 

Of  Canals.  43' 

§  420.  Company  between  the  circumstances  of  canals,  and  of 


421.  Examination  tf  the  proper  form  for  the  bed  of  a  canal 

at  its  union  v]th  the  source  whence  its  waters  are 
drawn.  \ 

422.  Investigation  of  the  i^?er  figure  of  banks  of  earth  used 

lor  conning  water. 

423.  Description  of^e  form  ouhe  bed  and  banks  of  a  canal. 

424.  Different  cases  n^hich  nav^ble  canals  are  used. 

425.  Relation  between  h,  area  o?KCana],  and  the  section  of 

the  vessels  that  nav^ate  it. 

426.  Description  of  a  lock. 

427.  Mode  of  passing  vessels  thrb»igh  a 


TABLE  OP  CONTENTS.  XX1X 

428.  Investigation  of  the  strength  of  the  walls  of  a  lock. 

429.  Description  of  the  gates  of  locks. 

430.  Investigation  of  the  proper  angle  for  the  leaves  of  the 

gates  of  locks. 

431 .  Principles  on  which  the  proper  height  of  locks  is  deter- 

mined. 

432.  Principles  that  determine  the  size  of  locks. 

433.  Defects  of  locks  placed  in  juxta-position. 

434.  Mode  of  diminishing  the  waste  of  water  in  deep  locks. 

435.  Estimate  of  the  waste  of  water  in  canals,  arising  from 

leakage  and  evaporation. 

436.  Feeders  for  canals. 

437.  Reservoirs  for  the  supply  of  canals. 

438.  Use  of  reservoirs  in  clarifying  the  water  of  canals. 

439.  Culverts. 

440.  Aqueducts. 

441.  Waste-gates. 

442.  Inclined  planes. 

443.  Description  of  the  lock  of  Betancourt. 

CHAP.    VIII. 

Of  the  Percussion  and  Resistance  of  Fluids.  459 

§  444.  Definition  of  the  Resistance  of  a  Fluid. 

445.  Theory  of  the  resistance  of  fluids. 

446.  Experimental  inferences  of  Bossut.      Analysis  of  the 

circumstances  of  the  percussion  of  fluids. 

447.  Circumstances  of  the  resistance  to  bodies  immersed  in 

fluids. 

448.  Circumstances  of  the  resistance  to  bodies  floating  at  the 

surface  of  liquids. 

449.  Limit  to  the  velocity  with  which  bodies  can  move  at  the 

surface  of  liquids. 

450.  Formulae  for  the  resistance  of  fluids  by  Bossut,   and 

Romme. 

CHAP.  IX. 

Of  the  Motion  of  Waves.  467 

&  451.  Mode  in  which  Waves  may  be  formed  at  the  surface  of  a 
liquid. 

452.  Comparison  between  the  motion  of  waves  and  the  oscil- 

lation of  a  liquid  in  a  bent  tube. 

453.  Theory  of  the  oscillations  of  a  liquid  in  a  bent  tube. 

454.  Application  of  the  foregoing  theory  to  the  case  of  waves. 

455.  Reflection  of  waves. 

456.  Interference  of  systems  of  waves. 

457.  Propagation  of  waves  through  vertical  orifices. 

458.  Examination  of  the  phenomena  of  waves  raised  by  the 

winds. 

459.  Effect  of  inclined  obstacles  upon  the  motion  of  waves. 

1 


TABLE  OF  CONTENTS. 


CHAP.  X. 


Of  the  Motion  of  Elastic  Fluids.  474 

§  460.  Theory  of  the  Motion  of  Elastic  Fluids. 

461.  Determination  of  the  velocity  with  which  atmospheric 

air  rushes  into  a  vacuum. 

462.  Velocity  of  air  of  different  densities  entering  spaces  con- 

taining air. 

463.  Velocities  of  gases  other  than  atmospheric  air. 

464.  Effect  of  the  air's  elasticity  on  its  motion. 

465.  Influence  of  adjutages  on  the  motion  of  air. 

466.  Effect  of  the  lateral  communication  of  motion  in  gases. 

467.  Motions  of  air  produced  by  variations  of  temperature. 

468.  Tables  of  the  velocity  of  steam. 

CHAP.  XI. 

Of  the  Motion  of  Gases  in  Pipes.  479 

§  469.  Theory  of  the  motion  of  air  in  Pipes. 
470.  Applications  of  the  theory. 

CHAP.  xn. 

Of  the  Motion  of  Air  in  Chimnies.  483 

§  471.  Theory  of  the  motion  of  air  in  chimnies,  abstracting  the 
resistance. 

472.  Effects  of  friction  in  general,  and  in  flues  of  different  sub- 

stances. 

473.  Reason  of  the  difference  in  the  friction  of  air  against 

different  substances. 

CHAP.  XII. 

Of  the  Winds.  486 

§  474.  Definition  and  general  cause  of  the  Winds  ;  causes  of 
the  diversities  of  temperature  at  different  points  on  the 
surface  of  the  earth. 

475.  Effect  of  elevation    on    the    temperature    of  places. 

Formula  that  represents  the  circumstances  that  influ- 
ence climate. 

476.  Influence  of  the  nature  of  the  surface  on  climate. 

477.  Temperature  of  lakes-  and  morasses  ;  effects  of  cultiva- 

tion. 

478.  Recapitulation  of  the  circumstances  that  influence  the 

temperature. 

479.  Manner  in  which  the  temperature  and  motion  of  the  air 

is  affected  by  the  temperature  of  the  surface,  and  the 
rotary  motion  of  the  earth. 

480.  Classification  of  the  winds. 

481.  Phenomena  of  the  Trade  Winds. 

482.  Seat  and  character  of  the  Variables. 

483.  Phenomena  of  the  Monsoons. 

484.  Local  modifications  of  the  trade  winds  and  monsoons. 


TABLE  OF  CONTENTS.  XXXI 

485.  Westerly  winds  between  the  parallels  of  30°  and  40°. 

486.  Variable  winds  of  the  temperate  and  frigid  zones. 

487.  Land  and  Sea  Breezes. 

CHAP.  XIV. 

Of  the  Motion  of  Vapour  in  the  Atmosphere.  502 

§  488.  Cause  of  the  existence  of  Vapour  in  the  atmosphere. 

489.  Phenomena  of  an  atmosphere  of  vapour  over  a  sphere  of 

uniform  temperature  and  surface. 

490.  Phenomena  in  the  case  of  an  uniform  decrease  of  tem- 

perature from  the  poles  to  the  equator. 

491.  Influence  of  the  aerial  part  of  the  atmosphere. 

492.  Effects  of  diminution  of  temperature  in  rising  in  the  aerial 

atmosphere. 

493.  Motions  in  the  aqueous  atmosphere,  growing  out  of  the 

unequal  distribution  of  land  and  water. 

494.  Influence  of  the  winds  on  the  distribution  of  vapour. 

495.  Origin  of  springs  and  rivers. 


CORRIGENDA. 


On  PAGE  3       line  22,  for  "  cos.2  a+cos.2  fc+eos.2  c=0,"  read 

"  cos.2  a+cos.2  fc+  cos.2  c  =  l." 

14     line  12,  for  "  cos.2  a+cos.2  6+cos.2  c=0,"  read 
"cos.2  a+cos.2  6+cos.2  c  =  l." 

42     line  24,  for  "  *=•—  —  +-7—  ~77  ,"  read 

<U  -  u         7J  -  V 

<ff  If 

u/  --  x  -     » 

—          *"- 


54     line  31,  /or  "g-"  read  "i/." 
112     line  1,    for  "CHAP,  v"  read  "CHAP,  iv." 

Wd  Wd 

160     line  7,    /or  "P=—  ,"  read  "  P  =  -Q-  • 


Wdb 
line  12,  /or  "P=-^-  ,»  read  "P=-^-   » 

270     (291)      for  "1  :  1+jfc,"  read  "l+i  h  :  1." 
320     (355)     for  "  fc=j^  ,"  read  "/i=^-." 

327  line  10,  for  "substances,"  read  "surfaces." 

333  line  13,  Jill  in  the  blank,  "  (       ),"  with  "  (299)." 

422  line  31,  for  "100,"  read  "av." 

449  line  4,    for  "friction,"  read  "function." 

437  line  53,  for  "level,"  read  "lever." 


us..  A  remains,      < 


BOOK  I. 

OF  EQUILIBRIUM. 

CHAPTER  I. 

GENERAL  PRINCIPLES. 

1.  MECHANICS  is  the  department  of  Physical  Science  which 
treats  of  Motion,  and  of  the  construction  of  Machines. 

2.  Bodies  appear  to  us  to  be  in  motion,  when  they  change  their 
position  in  respect  to  other  bodies  that  we  conceive  to  be  at  rest  ; 
but  even  bodies  that  appear  to  our  earlier  investigation  to  be  ab- 
solutely at  rest,  may  be  in  a  state  of  rapid  motion,  as,  for  instance, 
the  body  of  the  earth  itself;  which,  to  the  uneducated  and  igno- 
rant, appears  solid  and  immovable,  although  it  can  be  shown  by 
scientific  proofs  to  be  a  state  of  rapid  motion,  both  of  rotation 
and  translation.     Hence  we  refer  motion,  in  the  abstract,  to  un- 
bounded and  immovable  space.     As  space  is  extended  in  three 
dimensions,  a  body  may  move  in  the  direction  of  any  one  of  them, 
or  may  have  a  motion  intermediate  between  two  or  more  of  them  : 
it  may  rise  or  fall,  approach  or  recede,  pass  to  the  right  or  to  the 
left,  or  its  motion  may  be  combined  of  two,  or  all  three  of  those 
varieties. 

In  order  to  render  these  circumstances  of  motion  definite,  we 
refer  the  position  of  a  point  to  three  planes,  supposed  to  be  im- 
movable in  absolute  space,  and  which  cut  each  other  at  right 
angles. 

3.  The  perpendicular  distances  from  the  point  to  these  three 
planes,  are  called  its  Co-ordinates;  the  mutual  intersection  of  any 
two  of  the  planes  is  called  an  Axis,  and  the  common,  intersection  of 
three  planes  is  called  the  Origin  of  the  co-ordinates.     Each  of 
the  axes  is  parallel  to  one  of  the  co-ordinates,  for  the  common 
intersection  of  two  of  the  planes  is  perpendicular  to  the  third, 

1 


2  OP    EQUILIBRIUM.  \JBook  L 

and  the  co-ordinates  are  by  definition  each  drawn  perpendicular 
to  one  of  the  planes. 

4.  If  the  point  be  in  motion,  it  will  be  shown  by  the  change  in 
the  length  of  one  or  more  of  the  co-ordinates,  and  of  the  positions 
of  the  points  in  which  the  co-ordinates  cut  the  planes. 

5.  The  cause  by  which  a  body  is  set  in  motion,  whatever  be 
its  nature,  is  called  a  Force.     Forces  may  be  of  various  descrip- 
tions, but  as  they  all  produce  motion,   they  may  be  compared 
with  each  other,  and  made  commensurable  by  means  of  the  mo- 
tion they  produce  ;  and  we  judge  of  the  intensity  of  a  force  by 
the  quantity  of  motion  it  is  capable  of  causing.     As  we  know 
nothing  of  forces,  except  by  their  effects,  we  may  hence  assume 
that  the  force  is  always  measured   by  the  quantity  of  motion  it 
impresses  upon  a  point,  and  the  latter  is  always  proportioned   to 
the  velocity  of  this  point. 

6.  For  the  more  convenient  comparison  of  forces,  we  measure 
them  in  terms  of  some  conventional  force,  taken  as  the  unit.    Of 
this  we  have  a  practical  illustration  in  the  manner  in  which  the 
forces  of  steam-engines  and  water-wheels  are  compared,  in  terms 
of  the  conventional   force  called  a  horse-power.      The   intensity 
of  forces  estimated  in  terms  of  some  conventional  unit,  may  then 
be  denoted   by  numbers,  expressed   algebraically  by  letters,  or 
represented  by  lines  of  definite  magnitude. 

7.  The  circumstances  which  must  be  known   in  respect  to  a 
force,  besides  its  intensity,  are — the  place  where  it  nets,  or,  its 
point  of  application,  and  its  direction.      The  point  of  application 
is  defined  in  the  mode  we  have  already  explained,   by  referring 
it  by   means  of  co-ordinates,  to  three  rectangular  planes.      The 
direction  of  a  force  is  that  in  which  it  tends  to  cause  a  point  to 
move  :   it  is  usually  represented  by  a  straight  line  drawn  in  that 
direction  from  the  point  of  application  ;  and  if  upon  this  line  be 
set  off  the  number  of  units  from  a  scale,   which  corresponds  to 
the    measure  in  terms  of  the  conventional    force,  used   as    the 
means  of  comparison,   the  force  will  be  represented  by  it,  both 
in  magnitude  and  direction.     The  direction   will  be  defined  in 
respect  to  the  three  co-ordinates  of  the  point  of  application,   by 
means  of  the  three  angles  which  it  makes  with  these  three  lines. 

In  order  to  give  the  method  all  the  extension  of  which  it  is 
capable,  these  angles  must  be  estimated  of  all  magnitudes,  from 
0°  to  180°.  The  co-ordinates,  by  this  method,  need  not  be  con- 
ceived to  be  produced  beyond  the  point  of  application  ;  and  when  in 
calculation  we  employ  the  angular  functions,  those  which  have 
different  algebraic  signs  in  the  first  and  second  quadrants,  will 


Book  /.]  OP    EQUILIBRIUM.  3 

be  best  suited  to  express  the  position  in  which  the  line  of  direc- 
tion lies.  Thus  when  the  cosine  is  used,  and  its  algebraic  sign 
is  positive,  the  line  will  lie  on  the  side  of  the  point  of  applica- 
tion towards  the  plane  to  which  the  co-ordinate  is  perpendicular ; 
and  when  it  is  negative,  it  will  be  turned  towards  the  opposite 
direction. 

Between  the  cosines  of  the  three  angles  the  direction  of  a  force 
makes  with  the  co-ordinates  of  its  point  of  application,  there  is  a 
constant  relation  which  may  be  thus  expressed  : 
cos.2a+cos.26+cos.2c  =  l; 

for  the  line  of  direction  will  be  the  diagonal  of  a  right-angled  par- 
all  el  opipedon,  whose  sides  are  in  the  direction  of  the  three  co- 
ordinates, and  the  parts  cut  off  from  the  latter  will  respectively 
represent  the  cosines  of  the  angles  they  make  with  the  line  of  di- 
rection, the  latter  being  the  radius ;  now  as  the  square  of  the 
diagonal  of  a  right  angled  parallelepiped  is  equal  to  the  sum  of 
the  squares  of  its  three  sides,  the  square  of  radius,  or  unity,  is 
equal  to  the  sum  of  the  squares  of  the  three  cosines. 

If  the  line  of  direction  lies  in  the  same  plane  with  the  two  co- 
ordinates, with  which  it  makes  the  angles  a  and  6,  the  angle  c  be- 
comes a  right  angle,  and  its  cosine  =0,  hence,  in  this  case, 
cos.2o-j-  cos.26=0. 

When  all  the  forces  that  are  under  consideration  are  parallel 
to  each  other,  one  of  the  axes  may  be  so  taken  as  to  be  parallel 
to  their  direction ;  two  of  the  angles  in  this  case  become  right  an- 
gles, and  the  equation  becomes 

cos.2a=l. 

8.  When  more  than  a  single  force  acts  upon  a  body,  it  is  ob- 
vious that  it  will  riot  move  in  a  direction,  or  with  an  intensity 
that  is  due  to  one  of  the  forces  alone,  but  will  be  influenced  by 
all  the  forces  collectively.  Hence,  when  a  number  of  forces  act 
upon  the  same  body,  they  respectively  modify  each  other,  and 
may  under  certain  circumstances  completely  neutralize  each  other. 
When  forces  thus  destroy  each  other's  action,  equilibrium  is  said 
to  exist  among  them,  and  the  body  on  which  they  act  is  said 
to  be  in  equilibrio,  under  their  joint  action.  It  has  been  found 
most  easy  to  deduce  the  expressions  which  denote  the  motion  of 
a  body,  from  those  which  denote  the  conditions  of  equilibrium; 
hence  it  becomes  necessary  that  the  conditions,  under  which 
forces  produce  equilibrium,  should  be  first  investigated. 


4  OP    EQUILIBRIUM.  [Book  I. 

CHAPTER  II. 

EQUILIBRIUM  OP  FORCES  ACTING  IN  THE  SAME  LINE. 

9.  THE  simplest  case  of  equilibrium  is  when  two  forces  act  in  the 
same  line,  with  equal  intensities,  but  in  contrary  directions.   We 
represent  this  contrariety  of  direction  by  means  of  the  Alge- 
braic signs  +  and  — .     When  more  than  two  forces  act  in  the 
same  line,  it  is  obvious  that  equilibrium  can  only  exist  when  the 
joint  intensities  of  those  that  act  in  one  direction,  are  exactly 
equal  to  the  joint  intensities  of  those  which  act  in  opposition   to 
them.     Expressing  this  difference  of  direction,  by  considering 
one  set  of  forces  as  negative  in  respect  to  the  other,  we  obtain  the 
algebraic  expression 

A+B+C+&c.=0, 
or  in  words. 

Equilibrium  exists  among  a  number  of  forces  acting  in  the 
same  straight  line  when  their  sum  is  equal  to  0. 

10.  When  a  number  of  forces  acting  upon  a  body  are  not  inequi- 
librio,  we  may,  without  altering  the  circumstances  under  which 
the  body  is  placed,  conceive  their  united  action  to  be  replaced  by 
that  of  a  single  force,  under  which  the  body  would  move  exactly, 
as  if  the  whole  continued  to  act.     A  force  which  thus  produces 
the  same  effect  as  a  number  of  others,  and  may  therefore  identi- 
cally replace  them,  is  called  their  Resultant;  the  several  forces 
whose  action  it  thus  identically  replaces  are  called  its  Compo- 
nents. 

11.  If  a  force  equal  in  magnitude  to  the  Resultant,  but  contrary 
in  direction,  be  applied  to  the  point  at  which  the  latter  would 
act  if  its  components  were  removed,  it  will  be  obviously  in  ex- 
act equilibrium  with  the  components;  for  the  case  becomes  that 
of  two  equal  for^s  acting  in  the  same  straight  line,  but  in  con- 
trary directions.     Hence  if  any  number  of  forces  be  in  equilibrio, 
any  one  of  them  must  be  equal  in  magnitude,  and  contrary  in  di- 
rection to  the  resultant  of  all  the  rest.     If,  therefore,  we  have  the 
relations  that  exist  between  the  Resultant  and  its  components, 
we  can  thence  deduce  the  conditions  of  equilibrium  of  any  forces 
whatsoever. 

These  relations  may  for  convenience  of  investigation  be  divided 
into  three  cases,  those  of : 


Book  /.]  or  EQUILIBRIUM.  5 

1.  Forces  converging  to  a  point ; 

2.  Parallel  forces ; 

3.  Forces  acting  in  one  plane,  but  neither  parallel  nor  con- 
verging to  a  single  point. 

In  considering  the  latter  case,  we  shall  have  occasion  to  speak 
of  the  conditions  of  equilibrium  of  forces  acting  in  any  direction 
whatsoever,  but  all  our  applications  can  be  referred  to  the  case 
of  their  being  situated  in  one  plane. 


OF    EQUILIBRIUM.  \Book  I. 


CHAPTER  III. 

EQUILIBRIUM  OF  FORCES  CONVERGING  TO  A  POINT.     COMPO- 
SITION AND  RESOLUTION  OF  FORCES. 

12.  The  Resultant  of  the  two  forces  converging  to  a  point,  is 
represented  both  in  magnitude  and  direction,  by  the  diagonal  of 
a  parallelogram  constructed  on  the  two  forces  as  sides. 

First : — Let  the  directions  of  the  two  forces  be  at  right  angles 
to  each  other,  and  call  the  one  X,  and  the  other  Y.  Let  R  be 
the  unknown  magnitude  of  the  resultant,  and  a  the  angle  which 
it  makes  with  the  direction  of  X.  If  we  suppose  the  two  forces 
to  be  extremely  small  and  represented  by  their  differentials  rfX 
and  dY,  and  that  they  vary  according  to  the  same  law,  so  that 
when  rfX  becomes  successively  2dX,  3dX,  &c.  dY  becomes 
2dY,  3dY,  &c. ;  it  will  be  evident  that  the  angle  a  will  not  vary, 
and  that  the  resultant  will  be  constant  in  its  direction  ;  its  increase 
will  also  follow  the  same  law  with  its  components,  and  if  repre- 
sented at  first  by  dR,  it  will  become  in  similar  succession  2dR, 
3dR,  &c.  Thus  in  the  successive  increments  of  the  three  forces, 

the  ratios  between  the  resultant  R  and  the  two  components  X  and 

X 

Y  will  remain  constant.     The  relation  -=—  being  constant,  may 

1%- 
be  represented  by  a  function  of  the  constant  angle  a.     This  fact 
may  be  expressed  by  the  notation 

g=9(«)-  (1) 

But  as  the  angle  comprised  between  Y  and  R  is  also  constant, 
the  relation  between  these  two  quantities  may  be  represented  by 
some  function  of  it ;  and  this  function  will  obviously  be  of  the 
same  form  with  that  which  represents  the  former  relation,  or  to 
express  it  algebraically,  the  angle  being  the  complement  of  a 

!=<p(90'_  a).  (2) 


Book  I.] 


OF  EQUILIBRIUM. 


In  order  to  render  the  rest  of  the  investigation  more  obvious, 
we  must  have  recourse  to  the  anriexed  figure. 


In  this,  the  rectangular  forces  X  and  Y  are  represented  in  magni- 
tude and  direction  with  the  undetermined  resultant  R  lying  be- 
tween them,  for  its  direction  must  of  necessity  be  intermediate, 
and  in  the  same  plane  with  them.  Let  us  now  consider  the  force 
X  as  the  resultant  of  two  others,  the  first  of  which  x  is  in  the  di- 
rection of  the  force  R,  and  the  second  x'  is  at  right  angles  to  it ; 
the  angle  comprised  between  the  directions  of  X  and  x  is  the 
same  with  that  contained  between  X  and  R,  or  is  equal  to  a,  and 
the  angle  contained  between  X  and  x'  is  90° —  a.  The  same  re- 
lation will  then  exist  between  these  three  forces,  taken  by  pairs, 
that  exists  between  X,  Y,  and  R,  or 


(3) 


-=9(90°—  a) 


But  from  the  equations  (1)  and  (2)  we  have,  multiplying  by  R, 
X=R.<p(«),    Y=R.9(90°—  a),  (4) 

and  from  the  equations  (3)  we  obtain  in  like  manner 

ar=X.<p(«),    #'=X.9(90°—  a),  (5) 

substituting  the  values  of  9  (a),  and  9(90°—  a)  from  equations  (1) 

and  (2),  we  obtain 


_         _ 

*~R  '    x~  R   ' 

Resolving  Y  in  a  similar  manner  into  the  two  rectangular  com- 
ponents y  and  ?/',  one  of  which  y  is  in  the  direction  of  R,  we  ob- 
tain by  a  similar  operation 

XY 


OF  EQUILIBRIUM.  [Book  I. 

The  force  R  being  the  resultant  of  the  two,  X  and  Y,  is  also  the 
resultant  of  their  four  components  x,  y,  x',  y',  whose  values  are 
X2     Y2    XY     XY 

IT'  IT'  IT'  IT' 

but  the  two  last  x'  and  y',  are  equal  in  magnitude,  and  because 
they  respectively  make  right  angles  with  R  on  its  opposite  sides  ; 
they  act  in  the  same  line  in  contrary  directions  ;  hence  they  mu- 
tually destroy  each  other's  actions,  and  the  resultant  R  is  made 
up  of  the  two  remaining  forces,  which  act  in  the  same  direction 
with  it,  or 


whence 

R=v/(X2+Y2),  (6) 

which  is  the  expression  for  the  magnitude  of  the  diagonal  of  a 
right  angled  parallelogram  whose  sides  are  X  and  Y  ;  therefore 
the  resultant  of  two  rectangular  forces  is  represented  in  magni- 
tude by  the  diagonal  of  the  parallelogram  constructed  on  the  two 
forces  as  sides. 

That  it  is  also  represented  by  it  in  direction  will  be  obvious 
from  a  few  simple  considerations. 

The  value  of  R  being  thus  determined,  call  the  angle  which 

the  diagonal  of  the  parallelogram  makes  with  the  side  X,  6,  then 

X=R.  cos.  6;  (7) 

substituting  the  value  of  X  from  the  equation  (4)  and  dividing 

by  R 

cos.  b=cp.(a). 

The  unknown  function  of  the  angle  a  may  therefore  be  always 
represented  by  the  cosine  of  the  known  angle  6  ;  and  if  there  be 
any  case  in  which  a  =  6,  the  equality  must  hold  good  in  all  others. 
Now  if  the  forces  X  and  Y  be  equal,  the  resultant  R  must  be  equi- 
distant in  direction  from  the  directions  of  the  two  forces,  and  the 
angle  a  will  become  the  angle  which  the  diagonal  of  the  paral- 
lelogram makes  with  the  side  X,  or  a=b  ;  therefore  the  resultant 
of  two  rectangular  forces  is  not  only  represented  in  magnitude, 
but  in  direction,  by  the  diagonal  of  the  parallelogram  constructed 
upon  the  two  forces  as  sides. 

Next  suppose  that  the  two  forces,  X  and  Y,  are  not  rectangu- 
lar, but  make  with  each  other  any  other  angle  a.  Resolve  X  in 
two  other  forces,  one  of  which,  #,  is  in  the  direction  of  Y,  the 
other  x  perpendicular  thereto  ;  the  resultant  will  therefore  be  the 
resultant  of  the  three  forces  Y,  x,  and  x',  but  as  Y  and  x  are  in 
the  same  direction,  they  have  a  resultant  which  is  equal  to  their 
sum,  and  R  becomes  the  resultant  of  two  rectangular  forces, 
whose  magnitudes  are  respectively  Y+#,  and  #',  and  from  (6) 


Book  /.] 


OF  EQUILIBRIUM. 


or 


but 


(8) 


(9) 


JL2=3?+x'2,  and  a?=X  cos.  a, 
substituting  these  values  in  the  equation,  (8), 

R2=X2+2XY  cos.  a+Y2, 
which  shows  that  the  resultant  R  is  represented  in  magnitude  by 
the  diagonal  of  the  parallelogram  constructed  upon  the  forces  X 
and  Y  as  sides.  Also  is  it  represented  in  direction,  for  the  diago- 
nal of  the  parallelogram  constructed  on  Y-f-#  and  x'  as  sides,  is 
identical  not  only  in  magnitude,  but  in  position  with  the  diagonal 
of  the  parallelogram  constructed  on  X  and  Y. 

The  angle,  </,  may  vary  between  0°  and  180° ;  when  it  ex- 
ceeds a  right  angle,  the  quantity,  x,  becomes  negative  in  respect 
to  Y ;  the  quantity,  cos.  a,  also  becomes  negative,  and  the  second 
term  of  the  equation  is  negative. 

In  order  to  render  this  part  of  the  investigation  more  clear,  we 
annex  a  construction,  in  the  two  cases  of  obtuse  and  acute  angles, 
contained  between  the  two  forces. 


13.  The  problems  that  have  relation  to  finding  the  resultant  of 
two  forces,  when  the  components  are  given,  and  to  finding  the 
components  when  the  resultant  is  given,  go  by  the  name  of 
"  The  Composition  and  Resolution  of  Forces."  All  the  questions 
in  which  forces  are  to  be  composed  or  resolved,  may  be  solved  by 
means  of  plane  trigonometry ;  and  in  general,  of  two  forces,  their 
resultant,  and  the  three  angles  they  respectively  make  with  each 
other,  any  two  being  given,  the  remainder  may  be  found. 

(1).  When  the  two  components  and  the  angle  they  contain  are 
given,  we  have  from  the  equation,  (9), 

R=  ^  (Xa+2XY  cos.  a+Y3)  ;  (10) 

2 


10  Or  EQUILIBRIUM.  [Book  L 

when  the  angle,  a,  is  a  right  angle, 


X=R  cos.  a,  Y=R  sin.  a. 
(2).  In  the  figure  beneath,  the  force  X,  the  resultant  R,  and  a 
line  equal  and  parallel  to  Y,  make  up  a  triangle,  of  which  the 
angles  are  :  the  supplement  of  a  ;  the  angle  6,  equal  to  the  angle 
contained  by  the  force  Y  and  the  resultant  R  ;  and  the  angle  c, 
contained  by  the  resultant  and  the  force  X  ;  hence  as  the  sides  of 
triangles  are  proportioned  to  the  sines  of  the  opposite  angles, 


R  :  X  :  Y  :  :  sin.  a  :  sin.  6  :  sin.  c,  (12) 

and  if  the  directions  of  X  and  Y  are  rectangular, 

R=-^— =-Z-r,  (is) 

cos.  c        cos.  b 
X=R  cos.  c,    Y=R  cos.  6.  (14) 

14.  Three  forces  converging  to  a  point  are  in  equilibrio  when 
each  is  proportioned  to  the  sine  of  the  angle  contained  by  the  di- 
rections of  the  other  two  ;  they  are  also  in  equilibrio  when  rep- 
resented by  the  three  sides  of  a  plane  triangle  ;  and  hence  the  di- 
rections of  the  three  forces  must  lie  in  the  same  plane,  and  the 
sum  of  any  two  must  be  greater  than  the  third. 


Book  /.]  OF   EQUILIBRIUM.  11 

Let  the  three  forces  be  X,  Y  and  Z,  (by  §  10),  the  force  Z 


will  be  equal  to  the  resultant  of  X  and  Y  and  contrary  in  direc- 
tion, hence  the  relation  between  their  magnitudes  may  be  ex- 
pressed, by  substituting  Z  forR  in  analogy  (12),  as  follows: 

Z  :  X  :  Y  :  :  sin.  a  :  sin.  6  :  sin.  c  ;  (15) 

but  the  angle  a,  is  the  angle  contained  by  the  forces  X  and  Y  ;  the 
angle  6,  is  the  supplement  of  the  angle  b'  contained  by  the  direc- 
tions of  Z  and  Y  ;  and  the  angle  c  is  the  supplement  of  the  angle 
c  contained  between  the  directions  of  Z  and  X  ;  and  as  the  sines 
of  angles  and  their  supplements  have  the  same  magnitude, 

Z  :  X  :  Y  :  :  sin.  a  :  sin.  6'  :  sin.  c  ;  (16) 

or  the  forces  are  each  proportioned  to  the  sines  of  the  angles  con- 
tained by  the  directions  of  the  other*  two.  As  the  forces  X  and 
Y,  with  their  resultant  R,  are  represented  in  magnitude  by  the 
three  sides  of  a  triangle,  so,  as  Z=R, 
the  three  forces  X,  Y  and  Z  are  also  rep- 
resented in  magnitude  by  the  sides  of  a 
triangle.  This  triangle  may  be  formed  by 
drawing,  through  the  extremity  of  the  line 
representing  one  of  the  forces  X,  a  line 
equal  and  parallel  to  Y,  and  joining  the 
ends  of  the  last  line  to  the  point  at  which 
the  forces  act.  This  last  line  is  obvious- 
ly equal  to  Z,  and  in  the  same  direction 
produced,  for  it  is  the  diagonal  of  the  par- 
\_.  allelogram,  of  which  X  and  Y  are  sides. 


12 


OP   EQUILIBRIUM. 


[Book  I. 

A  triangle  whose  sides  are  proportioned  to  the  forces,  and  per- 
pendicular to  their  direction,  may  be  formed  as  in  the  figure  be- 
neath, by  drawing  perpendiculars  from  points  taken  at  will  in  the 

Z 


directions  of  the  forces.  It  will  be  manifest  that  the  three  angles 
a',  &',  c',  of  this  triangle  are  respectively  the  supplements  of  the 
three  angles  a,  6,  c,  that  the  direction,  of  the  forces  make  with 
each  other ;  hence  the  triangle  thus  constructed  will  be  similar  to 
that  formed  in  the  former  construction,  whose  sides  represent 
the  magnitude  of  the  three  forces. 

Three  oblique  forces  cannot  be  in  equilibrio,  unless  their  di- 
rections converge  to  a  single  point,  for  the  resultant  of  any  two 
of  them  must  be  equal  and  opposite  to  the  direction  of  the  third  ; 
and  hence  its  direction  must  pass  through  the  point  to  which  the 
directions  of  the  others  converge. 

As  the  resultant  of  any  two  of  the  forces  lies  in  the  same  plane 
with  them,  being  the  diagonal  of  a  parallelogram  of  which  they 
are  the  sides,  the  third  force,  which  is  in  the  direction  of  this  re- 
sultant produced,  must  also  lie  in  the  same  plane.  The  three 
forces  that  are  in  equilibrio  being  represented  in  magnitude  by 
the  three  sides  of  a  triangle,  the  sum  of  any  two  of  which  must  be 
greater  than  the  third,  the  same  must  be  true  of  the  sum  of  the 
magnitudes  of  any  two  of  the  forces. 

15.  When  we  have  it  in  our  power  to  find  the  resultant  of  any 
two  forces,  we  may  proceed  to  find  the  resultant  of  three  or 
more ;  for  as  the  resultant  identically  replaces  its  components, 
we  may,  after  finding  the  resultant  of  any  two  of  the  forces,  con- 
ceive them  to  be  removed,  and  the  resultant  substituted  ;  the  re- 
sultant of  this  and  the  third  force,  will  be  the  resultant  of  the  three 
forces ;  and  this  last  resultant  may  be  again  combined  with  a 
fourth  force,  and  so  on. 

This  problem  may  be  illustrated  by  a  remarkable  geometric 
construction. 


Book  /.] 


OP    EQUILIBRIUM. 


13 


Let  AB,  AC,  AD,  AF,  AG,  represent  a  number  offerees  con- 
curring to  a  point ;  through  the  point  B,  draw  the  line  BH,  equal, 
and  parallel  to  AC  ;  through  the  point  H,  draw  HI,  equal  and 
parallel  to  AD  ;  through  the  point  I,  draw  IK,  equal  and  parallel 
to  AF  ;  and  through  the  point  J(,  draw  KL,  equal  and  parallel  to 
AG  ;  then  the  line  which  joins  L  to  A  ;  will  be  the  common  re- 
sultant. 

It  is  obvious  that  the  line  AH  is  the  resultant  of  AB  and  AC  ; 
AI  the  resultant  of  AH  and  AD,  or  of  AB,  AC,  and  AD  ;  the 
line  AK,  of  AI  and  AF,  or  of  AB,  AC,  AD  and  AF  ;  and  the  line 
AL,  of  AI  and  AG,  or  of  all  the  forces  AB,  AC,  AD,  AF,  and 
AG. 

This  construction  is  called  the  polygon  offerees.  If  the  poly- 
gon close,  or  the  last  line,  drawn  parallel  to  the  last  force,  end  at 
the  point  A,  the  forces  are  in  equilibrio,  and  the  resultant  is  equal 
toO. 

16.  The  resultant  of  three  rectangular  forces,  converging  to  a 
point,  is  represented  in  magnitude  and  in  direction,  by  the  diago- 
nal of  the  rectangular  parallelepiped,  whose  sides  represent  the 
three  forces  in  magnitude  and  direction. 

The  resultant  R',  of  two  of  the  forces  X  and  Y,  will  be  the  diago- 
nal of  the  rectangle  which  forms  one  of  the  faces  of  the  paral- 
lelepiped, of  which  these  forces  are  sides,  and 


14  OP    EQUILIBRIUM.  Book  /.] 

R'=x/(X2+Y2);' 

the  resultant  R,  of  R',  and  Z  will  be  the  diagonal  of  the  rectangle 
of  which  these  two  forces  are  sides,  and  which  will  cut  the  paral- 
lelepiped into  two  equal  and  similar  prisms  ;  it  will  therefore  be 
the  diagonal  of  the  parallelepiped,  and  will  be  represented  by  the 
formula 

R=  V  (R'2+Z2)  =  v/  (X3+Y2+Z3) .  (17) 

If  we  call  the  angle  the  direction  of  R  makes  with  X,  a ;  the 
angle  the  direction  of  R  makes  with  Y,  b  ;  the  angle  the  direc- 
tion of  R  makes  with  Z,  c  ;  these  angles  are,  as  has  been 
shown,  connected  by  the  following  relation, 

cos.2  o-f-cos.2  6-f-cos.2  c=0. 

We  may  find  the  values  of  X,  Y  and  Z,  when  the  force  R,  and 
the  angles  a,  6,  and  c,  are  given,  by  the  resolution  of  three  plane 
triangles,  of  each  of  which  the  hypothenuse  and  an  angle  are 
given,  thus : 

X=Rcos.  a,  Y=Rcos.  6,  Z=Rcos.  c.  (18) 

17.  In  those  investigations  in  mechanics  where  a  number  of 
forces  are  concerned,  it  is  usual  to  resolve  them  all  into  three  forces 
parallel  to  the  three  co-ordinates  ;  and  the  resultant  of  these  three 
sets  of  rectangular  forces  will  obviously  be  the  common  resultant 
of  all  the  forces.  The  formulae  given  above  (18)  furnish  a  con- 
venient mode  of  effecting  this. 

Call  the  several  forces  F,  F',  F",  &c.  the  angles  they  respec- 
tively make  with  the  co-ordinates  of  their  points  of  application  a, 
6,  c,  &c.,  a',  b',  c',  &c.,  a",  6",  c",  &c.  the  values  of  X,  Y  and 
Z,  in  (18)  become 

F  cos.  a,     F  cos.  6,     F  cos.  c,  &c. 

and  calling  the  three  rectangular  forces,  which  are  the  sum  of  the 
components  of  all  the  original  forces  X,  Y  and  Z, 

X=F  cos.  a-fF'  cos.  6+F"  cos.  c-h&c.    '  '*"' 

Y=F  cos.  a'+F'  cos.  &'+F"  cos.  c'-f  &c.     \          (19) 

Z=F  cos.  a"+F'  cos.  6"+F"  cos.  c"+&c.  j 

In  these  expressions,  X,  Y,  and  Z,  are  the  components  of  the 
resultant  of  all  the  forces  resolved  into  three,  mutually  at  right  an- 
gles to  each  other.  The  formulae  show  that  the  resultant  of  any 
number  of  forces,  resolved  into  a  component  in  any  given  direc- 
tion, is  equal  to  the  sum  of  the  components  of  all  the  forces,  re- 
solved into  directions  parallel  to  the  general  resultant. 

IS.  The  three  forces,  X,  Y,  and  Z,  are  not  situated  in  the  same 
plane,  and  hence  can  never  be  in  equilibrio,  so  long  as  any  one 
of  them  has  any  magnitude  ;  for  three  oblique  forces,  in  order  to 
be  in  equilibrio,  must  have  their  directions  in  one  plane ;  hence 
the  condition  of  equilibrium  among  any  number  of  forces,  each 


Book  /.]  O*    EQUILIBRIUM.  15 

resolved  into  three  at  right  angles  to  each  other,  and  parallel  to 
the  axes  of  the  co-ordinates,  becomes 

F  cos.  a+F'  cos.  6-fF" cos.  c+&c.=0   ~'<\\ 

F  cos.  a'+F'  cos.  6'+F"  cos.  c'+&c.=0     V  (20) 

F  cos.  a"-fF'  cos.  6"+F"  cos.  c"+&c.-=0  j   V  ' 

19.  There  is  another  case  in  which  forces  may  be  in  equilibrio, 
when  they  converge  to  a  point ;  this  happens  when  they  press 
the  point  against  a  surface,  which  opposes  a  resistance  to  the  mo- 
tion of  the  point  sufficient  to  prevent  its  penetrating  under  the 
action  of  the  forces.  This  resistance  will  be  exerted  in  a  direc- 
tion which  is  perpendicular,  or  a  normal,  to  the  surface;  for  were 
it  exerted  in  an  inclined  direction,  it  might  be  resolved  into  two 
components,  one  of  which  is  parallel,  the  other  perpendicular  to 
the  surface  ;  the  former  would  cause  the  point  to  move  along  the 
surface,  while  the  latter  alone  would  oppose  the  progress  of  the 
point ;  now  as  mere  resistance  can  never  generate  motion,  how- 
ever it  may  in  other  respects  affect  it,  the  resistance  of  the  sur- 
face must  be  exerted  in  the  direction  of  the  normal  alone. 

In  order  then  that  the  point  be  in  equilibrio,  it  is  no  longer  ne- 
cessary that  the  resultant  of  the  forces  shall  be  equal  to  0,  but 
merely  that  its  direction  shall  be  a  normal  to  the  surface  ;  this  it 
must  be,  otherwise  it  would  not  be  opposite  in  direction  to  the 
resistance  of  the  surface,  and  therefore  would  not  be  in  equilibrio 
with  it.  The  action  of  the  surface,  being  just  sufficient  to  keep 
the  point  in  equilibrio,  may  be  represented  by  a  force  equal  to 
the  resultant  of  all  the  other  forces,  but  acting  in  a  contrary  direc- 
tion, or  by  — R  ;  and  the  condition  of  equilibrium,  they  being  re- 
solved into  three,  parallel  to  their  co-ordinates,  will  be  (17) 

V(X2+Y2+Z2)— R=0,  (21) 

while  the  pressure  they  exert  upon  the  surface  will  be 

(22) 


16  OF    EQUILIBRIUM.  [Book  L 


CHAPTER  IV. 

IV.   EQUILIBRIUM  op  PARALLEL  FORCES.     CENTRE  OF  PAR- 
ALLEL FORCES. 

20.  The  conditions  of  equilibrium  among  parallel  forces,  may 
be  reached  by  steps  similar  to  those  employed  in  determining  the 
condition  of  equilibrium  among  converging  forces.     The  method 
of  finding  the  resultant  of  two  of  them  must  be  first  investigated  ;  a 
force  equal  and  opposite  to  this,  will  be  in  equilibrio  with  the  two 
first.     We  may  then  proceed  to  the  resultant  of  three,  four,  or 
any  number  of  forces;  and  a  force  equal,  and  opposite  to  it,  will 
cause  equilibrium  in  the  system,  and  the  relations  that  exist  among 
them  will  show  the  condition  of  equilibrium. 

Parallel  forces  cannot,  act  upon  a  single  point;  we  therefore 
suppose  the  points  on  which  they  act,  to  be  connected  in  some 
manner,  and  first,  by  an  inflexible  and  inextensible  line,  which  is 
called  their  line  of  application. 

There  is  one  case  in  which  parallel  forces  have  no  resultant. 
This  happens  when  there  are  but  two  that  act  on  opposite  sides 
of  the  line  of  application,  and  are  equal  in  magnitude;  or  when 
the  resultant  of  all  those  that  act  on  one  side  of  that  line,  is  ex- 
actly equal,  but  not  directly  opposite  to  the  resultant  of  all  those 
that  act  on  the  other  side.  Two  forces  thus  constituted,  are  called 
a  Couple.  It  will  be  obvious  that  their  effect  would  be  to  cause 
the  line  of  application  to  revolve,  and  were  it  free,  to  move 
around;  the  two  forces  would  finally  act  in  the  same  line,  and 
in  opposite  directions;  they  would  then  cease  to  be  parallel.  Two 
such  forces  then,  so  long  as  they  continue  parallel,  have  no  result* 
ant,  neither  can  they  be  in  equilibrio. 

21.  In  investigating  the  method  of  finding  the  resultant  of  par- 
allel forces,  it  will  be  seen  that  we  shall  consider  the  point  of  ap- 
plication of  a  force  to  be  changed.       A  force  will   obviously 
produce  the  same  effect,  whether  it  act  at  one  or  another  point  of 
the  same  inflexible  straight  line.    Of  this  we  have  a  physical  illus- 
tration in  the  fact,  that  when  a  simple  pressure  is  exerted  through 
the  intervention  of  a  rigid  bar,  it  is  wholly  unimportant  whether 
the  bar  be  long  or  short,  provided  its  weight  have  no  influence. 
We  may  therefore  conceive  the  point  of  application  of  a  force  to 
be  transferred  to  any  other  point  in  its  direction,  provided  we 
imagine  the  two  points  to  be  connected  by  an  inflexible  straight 
line. 

22.  The  resultant  of  two  parallel  forces  is  equal  to  their  sum, 


Book!.']  OF    EQUILIBRIUM.  17 

parallel  to  their  directions,  and  divides  their  line  of  application 
into  parts,  reciprocally  proportioned  to  the  magnitude  of  the  two 
forces. 

Suppose  first,  that,  as  in  the  figure,  two  converging  forces  act 
upon  the  same  side  of  the  line,  call  them  A  and  B  ;  their  action 


will  not  be  changed  by  supposing  their  respective  points  of  appli- 
cation to  be  transferred  to  the  point  C,  in  which  their  directions 
meet;  and  the  direction  of  their  resultant  must  therefore  pass 
through  this  point.  From  the  point  of  convergence,  then,  its 
point  of  application  may  be  transferred  to  the  point  d,  in  which  its 
direction  cuts  the  line  of  application,  and  this  will  therefore  be  the 
point  of  application  of  the  resultant.  Call  the  angle  at  which  the 
forces  are  inclined  to  each  other  p,  the  value  of  the  resultant  from 
(9)  is 

R=  </ A2+ AB  cos.  g +B2)  ;  (23) 

the  relation  of  the  forces  A  and  B  is  from  (12) 

A  :  B  :  :  sin.  /3  :  sin.  a,  (24) 

/3  being  the  angle  the  direction  of  B  makes  with  that  of  R,  and 
a  the  angle  that  the  direction  of  A  makes  with  that  of  R  ;  but  the 
ratio  of  the  sines  will  be  the  same  as  that  of  the  perpendiculars  a 
and  6,  let  fall  upon  the  respective  directions  of  the  forces,  from 
the  point  of  application  of  the  resultant :  hence 

A  :  B  :  :  b  :  a.  (25) 

We  shall  have  occasion  hereafter  to  recur  to  this  step. 

Now  the  equation  (23),  and  the  analogy  (25)  being  true,  what- 
ever be  the  magnitude  of  the  several  angles  the  forces  and  t  eir 
resultant  make  with  each  other,  will  be  true  when  the  lines  are 
parallel.  But  when  the  lines  are  parallel,  the  angle  £=180°  or 
=  0°  and 

cos.  £=1  ; 
3 


IS 


OF    EQUILIBBIUM. 


[Book  I. 


(26) 


hence  the  equation  (28)  becomes 

R=V(A2+AB+B2)and 
R=A-r-B 

or  the  resultant  is  equal  to  the  sum  of  the  forces. 

The  point  of  convergence  being  removed  to  an  infinite  distance, 
the  direction  of  the  resultant  becomes  parallel  to  the  directions 
of  the  two  forces  ;  and 


the  two  perpendiculars  a  and  b  become  proportioned  to  the  parts 
c  and  d  into  which  the  point  of  application  of  the  resultant  divides 
the  line  of  application  of  the  forces,  which  is  hence  divided  into 
parts  inversely  proportioned  to  the  two  forces. 

If  the  two  parallel  forces  act  on  opposite  sides  of  their  line  of 
application,  the  same  is  also  tnie  ;  but  in  this  case,  the  opposition 
of  their  direction  is  pointed  out,  by  one  of  the  forces  being  consi- 
dered negative  in  respect  to  the  other  ;  hence  their  algebraic  sum 
becomes  their  arithmetic  difference.  The  point  of  application  of 
the  resultant,  will  fall  in  the  prolongation  of  the  line  of  application, 
beyond  the  point  to  which  the  greater  force  is  applied ;  and  the 
parts  into  which  the  line  of  application  is  divided,  will  be  mea- 
sured from  the  point  of  application  of  the  resultant  to  those  of  the 
two  components. 

To  make  this  evident,  suppose  that  to  a  line  on  which  the  two 
forces,  A  and  B  are  applied,  a  third  force,  C,  is  also  applied  equal 
to  the  resultant,  and  opposite  in  direction;  this  force  will  be 
negative  in  respect  to  the  others,  and  will  keep  the  system  in  equi- 
librio ;  the  conditions  of  which  will  be  thus  expressed : 


A+B+C=0, 


(27) 


any  one  of  these  forces  then,  is  in  equilibrio  with  the  other  two ; 
and  the  resultant  of  these  two  would  be  equal  to  it,  and  opposite 


/.]  05»  EQUILIBRIUM.  19 

in  direction.  Let  the  two  forces,  whose  resultant  is  required,  be 
B  and  C  ;  the  resultant  being  equal  and  opposite  to  A,  will  have 
a  value,  which,  considering  that  one  of  the  two  forces  B  and  C, 
is  negative  in  respect  to  the  other,  may  be  thus  expressed  : 


and  as  the  point  of  application  of  C  divides  the  line  to  which  A 
and  B  are  applied,  into  parts  inversely  proportioned  to  the  two 
forces,  we  have 

A  :  B  :  :  c  :  d,       , 
and 

A+B  :B:  :  c+d  :  d  ; 
but 

A+B=C, 

and  c-\-d  is  the  whole  length  of  the  line  of  application,  measured 
from  the  point  to  which  A,  or  its  equal  and  opposite  force,  R  is 
applied  ;  hence  this  line  is  again  divided  into  parts  inversely  pro- 
portioned to  the  magnitudes  of  the  two  forces. 

23.  The  resultant  of  any  number  of  parallel  forces,  acting;  upon 
points  invariably  connected  with  each  other,  may  be  found  upon 
the  same   principle  by  which   ibe   resultant  of  any    number  of 
converging  forces  is  found.     Any  two  of  them  may  be  first  re- 
solved into  a  single  one  ;   this  may  be  combined  with  a  third,  and 
the  resultant  found  ;  the  second  resultant  may  be  combined  with 
a  fourth,  and  so  on.      In  this  way  it  will  be  found,  that  the  result- 
ant of  any  number  of  parallel  forces  is  equal  to  their  sum. 

Call  the  forces  A,  B,  C,  D,  &c. 
the  first  resultant  R'  being  equal  to  the  sum  of  A  and  B, 
R'=A+B, 

the  value  of  the  second  resultant  is 

R'=R'-fC=A+B-fC, 

and  of  the  last  resultant  — 

R=A+B+C+D+&c.  (28) 

In  this  expression,  as  in  the  others,  the  forces  that  act  in  oppo- 

site directions,  must  be  considered  as  positive  and  negative  in  re- 

spect to  each  other.     The  condition  of  equilibrium  is  obviously 

A-{-B  +  C  +  &c.=:0.  (29) 

24.  A  change  in  the  direction  of  two  parallel  forces,  provided 
they  tlo  not  cease  to  be  parallel,  produces  no  change  in  the  posi- 
tion of  the  point  to  which  their  resultant  is  applied  ;   but  the  re- 
sultant itself  will  have  its  direction  changed,  continuing  always 
parallel  to  its  component.      And  in  the  several  steps  by  which  the 
resultant  of  a  number  of  parallel  forces  is  found,  it  is  obvious  that 


20 


OP    EQUILIBRIUM. 


[Book  I. 


the  position  of  the  several  successive,  and  finally,  of  the  last  result- 
ant, remains  constant,  however  the  forces  vary  in  direction,  pro- 
vided they  do  not  cease  to  be  parallel.  Hence,  in  a  system  of 
parallel  forces,  if  the/several  forces  revolve  around  their  respec- 
tive points  of  application  without  ceasing  to  be  parallel,  the  re- 
sultant will  also  revolve  around  its  point  of  application,  always 
retaining  its  parallelism  to  its  components.  From  this  property, 
the  point  of  application  of  the  resultant  of  a  system  of  parallel  for- 
ces, is  called  their  Centre. 

25.  It  is  important  in  many  practical  cases,  to  be  able  to  deter- 
mine the  position  of  the  centre  of  a  given  system  of  parallel  forces, 
applied  to  points  forming  an  invariable,  or  rigid  system.  This  is 
effected  by  finding  the  value  of  its  co-ordinates,  in  terms  of  the 
several  forces,  and  their  respective  co-ordinates. 

Call  the  several  forces  A,  B,  C,  &c.,  their  respective  co-ordi- 
nates a,  a',  a'',  6,  6',  6'',  c,  c',  c"  &c.,  the  resultant  R,  and  its  co- 
ordinates X,  Y,  and  Z. 

In  the  following  figure,  let  A  •  and  B  represent  the  points  of 


application  of  two  of  the  forces  A  and  B,  and  R'  the  point  of 
application  of  their  resultant  R.  Let  O  be  the  origin  of  the 
co-ordinates  O  P,  O  N,  O  S,  and  a,  6,  and  x  the  co-ordinates 
of  the  two  forces,  and  their  resultant,  in  respect  to  the  plane  of 
0  P  and  O  N.  The  line  that  unites  the  points  in  which  the  co-or- 
dinates cut  this  plane,  is  a  straight  line,  for  the  three  forces  are  in 
the  same  plane.  Then  as  the  line  A  B  is  divided  by  the  result- 
ant into  parts  inversely  proportioned  to  A  and  B 

A  :  B  :  :  R'A  :  R'B ; 

and 

A-f  B  :  B  :  :  AB  :  R'B, 

or 


R':  B  :  :  AB  :  R'B. 


(30) 


Book  I.']  OP   EQUILIBRIUM.  21 

Draw  a  straight  line  through  the  point  B,  parallel  to  the  line  that 
joins  the  points  in  which  a,  6,  and  x  cut  the  plane  of  O  P  and  O  N. 
The  respective  distances  of  the  points  A  and  R  from  this  line 
will  be  a — 6,  and  x — 6. 

From  the  similarity  of  triangles 

AB  :  R'B  :  :  a— b  :  x— b  ; 
comparing  this  with  the  analogy  (30) 

R':  B  :  :  a — 6  :  x— b  ; 
whence 

R'(#— 6)=B(a — 6)  ; 

now  the  resultant  R'  is  equal  to  the  sum  of  the  two  forces  A  and 
B,  and  multiplying  these  equals  by  b  we  have 

R=A6+B6  ; 
adding  the  two  last  equations 

R'x=Aa-\-Eb.  (31) 

A  similar  chain  of  reasoning,  in  respect  to  each  of  the  other  two 
sets  of  co-ordinates,  gives 

(32) 
K^=Aa'+i56"  f    : 

by  division 

*=AoJBZ 

(33) 


K' 


Let  now  C  be  the  point  of  application  of  the  third  force  C  ;  R" 
the  point  of  application  of  the  resultant  of  the  two  forces  C  and  R7, 
of  which  the  latter  is  the  resultant  of  A  and  B.  A  similar  inves- 
tigation gives  for  the  values  of  the  co-ordinates  x  y'  z  of  the 
force  R" 


Rz+Cc" 
R+C       ; 

substituting  the  values  of  R'.r,  ~R'y  and  R'z, 

,_Aa+B6+Cc 

=  A+  B-f-  C 


'  A  +B  +  C 
Aa"+B6"+Cc' 
A  -f  B  -f  C 


OF    EQUILIBRIUM. 


[Book  I. 


It  is  now  obvious  that  the  same  method  may  be  extended  to 
any  number  of  forces  whatsoever,  and  that  we  should  finally  ob- 
tain for  the  values  of  the  co-ordinates  of  any  number  of  forces 
the  following  equations  : 


X~ 


A  -I-  B+C+&C. 
Ao'+By+C/+&c. 
A  +  B+C  +  &C. 
Aa"+Bfe"+Cc"+&c. 
A  +B+C  +&c. 
In  the  case  of  equilibrium  these  equations  become 

=0 


(34) 


A  H-B  +C  H-&C. 
Aa'+B6'+Cc+&c. 
A  +  B  +  C  +&c. 


=0 


Aa"+Bfe"-r-Cc"+&c 
A  +   B  +  C  +&c. 

The  expressions  (34)  may  be  made  to  assume  the  following 
form,  which  is  more  convenient  in  its  application  to  the  cases  that 
may  occur  in  practice  : 


~ 


Y=: 


F 

Fy 


2.  F 


(35) 


The  numerator  of  the  three  fractions  being  the  sum  of  the  pro- 
ducts of  the  respective  forces  into  their  distances  from  the  several 
planes,  and  2  .  F  being  the  sum  of  the  forces  themselves. 

26.  The  most  useful  applications  of  these  formulae,  are  to  the 
determination  of  the  centres  of  parallel  forces  in  lines,  surfaces, 
and  solids.  In  these  cases,  the  several  magnitudes  are  sup- 
posed to  be  divided  into  an  infinite  number  of  small  parts  or  ele- 
ments, each  of  which  is  acted  upon  by  an  equal  parallel  force. 
The  manner  in  which  the  foregoing  formulae  are  transformed 
into  others  adapted  to  this  research,  together  with  a  few  of  their 
applications  will  now  be  given.  It  will  however  be  obvious,  that 
these  formulae,  having  reference  to  the  small  elements  of  which 
the  magnitudes  are  made  up,  must  be  differential  ;  and  that  the 
discovery  of  the  position  of  the  centre  of  parallel  forces,  can  only 
be  effected  by  means  of  the  integral  calculus. 


Book  /.]  OP    EQUILIBRIUM.  23 

In  the  case  of  irregular  solids  and  lines  of  double  curvature,,  it 
is  necessary  to  refer  the  position  of  their  elements  to  three  rec- 
tangular planes  ;  and  hence  the  three  equations  for  the  co-ordinates 
of  the  centre  of  parallel  forces,  are  necessary.  In  the  case  of  sur- 
faces and  plane  curves,  no  more  than  two  of  these  equations  are 
necessary  ;  for  the  surface  itself,  or  the  plane  in  which  the  line  lies, 
may  be  taken  as  the  plane  passing  through  two  of  the  axes,  and  it 
will  only  be  necessary  to  define  the  position  in  respect  to  these 
two  axes. 

In  lines  that  are  symmetric  on  each  side  of  one  of  their  points,  in 
surfaces  that  are  symmetric  on  eachsideof  aline  thattraverses  them, 
and  in  solids  of  revolution,  but  one  of  the  equations  is  necessary  ; 
for  one  of  the  axes  may  in  the  first  instance  be  supposed  to  pass 
through  the  point  on  each  side  of  which  the  line  is  symmetric,  and  to 
be  perpendicular  to  the  line  at  that  point  ;  in  the  second  it  may  la 
assumed  as  coinciding  with  the  line  that  divides  the  surface  into 
two  equal  parts  ;  and  in  the  third,  it  maybe  taken  as  coinciding  with 
the  axis,  by  a  revolution  around  which  the  solid  is  described.  In 
all  these  cases,  the  centre  of  parallel  forces  will  be  situated  some- 
where in  the  common  intersection  of  the  two  planes,  its  distance 
from  which  is  therefore  =0,  or  if  these  be  the  planes  on  which  Y 
and  Z  fall, 

Y=0,  Z=0. 

Restricting  ourselves  to  cases  that  admit  of  the  use  of  no  more 
than  one  equation  (1).  In  symmetric  curves,  the  equation  (35) 

2  .Fx 
X~2TF 
becomes 


in  which  I  is  the  length  of  the  line,  and  x  the  variable  ordinate  of 
the  element  d/,  in  respect  to  the  assumed  plane. 

(2)  In  plane  surfaces  the  equation  becomes 

fxydx  (37) 

X-~T~  , 

where  s  is  the  area  of  the  surface,  whose  element  is  ydx. 

(3)  In  solids  of  revolution,  the  formula  for  the  position  of  their 
centre  of  parallel  forces  becomes,  if  being  the  ratio  of  the  circum- 
ference of  a  circle  to  its  diameter, 

_J  xfdx  (38) 


24  oy  EQUILIBRIUM.  [Book  I. 

EXAMPLES. 

(1)   To  find  the  centre  of  Parallel  Forces  of  a  circular  arc. 
Let  B  A  D  be  the  circular  arc,  whose  length  =1. 

A. 


Place  the  origin  of  the  co-ordinates  at  the  centre ;  make  the  ra- 
dius C  A=r,  CP=#,  mP=7/,  Am=s ; 
the  differential  equation  of  the  arc  s  is 

ds 
and 


whence 


x=r.  cos.- 


fxds=fxV(dx*+dy2}=r.2  sin.  —  +  C. 


When  the  arc  Am  becomes  equal  to  AD 


=-,  and  the  arbitrary  constant  C =0  ; 


I 


and  from  the  equation  (36) 
therefore 

/X=2r2sin.  ^;, 
Let  c  represent  the  chord  of  the  arc,  then 

c=2r.  sin.  7^  ; 
whence 

and 


ZX=rc, 


wherefore  the  distance  of  the  centre  of  gravity  of  a  circular  arc, 
from  the  centre  of  the  circle,  is  a  fourth  proportional  to  the  length 
of  the  arc,  the  chord,  and  the  radius. 

By  analogous  processes,  the  centres  of  parallel  forces  may  be 
found  in  other  curves,  that  are  symmetric  on  each  side  of  the  point 
in  which  the  axis  cuts  them,  thus : 


Book  /.]  OP    EQUILIBRIUM.  25 

The  centre  of  gravity  of  an  arc  of  a  cycloid,  that  is  divided  into 
two  equal  parts  by  the  diameter  of  the  generating  circle,  is  at  one 
third  of  the  perpendicular  height  of  the  arc,  from  the  vertex. 

In  a  semicircle  c=2r,  and  /=<TTT  hence 
err  :  2r  :  :  r  :  X  ; 


(2)    To  find  the  centre  of  parallel  forces  in  a  segment  of  a  circle. 

In  the  same  figure  let  the  radius  CA^r  and  let  the  part  of 
AC  intercepted  between  the  centre  and  the  chord  BD—  a;  let  the 
centre  of  the  circle  again  be  the  origin  of  the  co-ordinates  ;  the 
equation  of  the  circle  will  be 


whence  y=  \/  (r2  —  x2}  ; 

the  expression  (37)  gives 

«X=2/V(a*—  yz}xdx. 
Integrating  between  the  values  x=a,  andv# 


and  the  chord  c  =  2  V  (r2—  a2) 

whence 


Applying  analogous  methods,  we  obtain  the  following  results  : 
A  triangle  has  its  centre  of  parallel  forces,  in  the  line  drawn  from 

its  vertex  to  the  point,  that  bisects  its  base,  at  two  thirds  of  its 

length  from  the  vertex. 

In  a  trapezium,  two  of  whose  sides  are  parallel,  call  these 

two  sides  c  and  d,  and  the  straight  line  which  bisects  both,  a,  and 

let  the  origin  of  the  co-ordinates  be  in  the  point  where  this  line 

cuts  the  side  c  ; 

a      c+3d 


A  sector  of  a  circle,  has  its  centre  of  parallel  forces  in  the  radius 
that  divides  it  into  two  equal  parts  ;  and  its  distance  from  the  centre 
of  the  circle  is  a  fourth  proportional  to  the  arc,  the  chord,  and  two 
thirds  of  the  radius. 

The  distance  of  the  centre  of  parallel  forces  of  a  parabola,  from 
its  vertex,  is  equal  to  three  fifths  of  its  axis. 

The  centre  of  parallel  forces  in  a  cycloid,  is  in  the  diameter  of 
the  generating  circle,  at  the  distance  of  one  fourth  of  that  line  from 
the  vertex. 

In  the  surfaces  that  bound  solids  of  revolution,  if  being 
the  ratio  of  the  circumference  of  a  circle  to  its  diameter,  the 
element  of  the  surface  is  2«ydx,  hence  the  equation  (38)  becomes 


26  OP    EQUILIBRIUM.  \BoOk  I. 


~  f2<rfydx          s       » 

which  being  identical  with  (37,)  shows,  that  the  distance  of  the 
centre  of  parallel  forces,  in  the  surface  formed  by  the  revolution 
of  a  plane  curve  around  an  axis,  is  as  far  from  the  origin  of  the 
co-ordinates,  as  the  centre  of  parallel  forces  of  the  curve  by  whose 
revolution  it  is  generated. 

(3)  To  find  the  centre  of  parallel  forces,  of  a  solid  generated 
by  the  revolution  of  an  arc  of  an  ellipse,  or  of  the  segment  of  a 
spheroid. 

The  formula  (38)  is 

fsnfdx 
* 
the  equation  of  the  curve  is 


a  being  the  fixed  axis,  and  c  the  revolving  axis  of  the  spheroid  ; 
whence 

f(ax—  x2}xdx 
K~  f(ax—  x*)dx    • 

Integrating,  and  taking  the  vertex  for  the  origin  of  the  co-ordi- 
nates 

i  aa?  —  i  x4  _4o  —  3# 
X  =       2—  x*  =  60—  4x  X' 


When  the  segment  is  a  hemispheroid  tf=£a,  and  2#—  a,  which 
being  substituted  for  a, 

5 

X=8*; 

its  centre  of  parallel  forces  is  therefore  in  the  fixed  axis,  at  a  dis- 
tance of  |ths  of  its  length  from  the  vertex. 

These  expressions  being  independent  of  the  value  of  c,  are  true 
also  of  spheric  segments. 

In  a  hyperboloid  of  revolution 


In  a  solid  paraboloid,  the  centre  of  parallel  forces  is  at  the  dis- 
tance of  two-thirds  of  the  length  of  the  axis,  from  the  vertex. 

Geometric  methods  also  may  be  applied  to  the  discovery  of  the 
position  of  the  centre  of  parallel  forces.  Thus  in  the  case  of  a  tri- 
angle, it  may  be  shown  geometrically,  as  it  has  been  analytically, 
to  be  in  the  line  that  joins  the  vertex  to  the  point  which  bisects 
the  base,  at  the  distance  of  two  thirds  of  its  length  from  the  vertex. 


Book  /.]  OP   EQUILIBRIUM.  27 

In  the  triangle  ABC,  bisect  the  base  BC  in  E,  the  side  AB  in 
D,  draw  AE,  CD,  and  join  DE. 


The  surface  being  divided  into  two  equal  parts  by  the  line  AE, 
the  centre  of  parallel  forces  will  lie  in  this  line  ;  for  the  same  rea- 
son it  lies  in  the  line  CD,  and  must  therefore  be  in  the  point  g 
where  they  intersect  each  other. 
By  the  similarity  of  triangles, 

gD  :  gC  :  :  g-E  :  gA, 
g-E  :  gA.  :  :  DE  :  AC, 
DE  :  AC  :  :  AB  :  DB, 
g-E  :  gA  :  :  AB  .  DB  ; 
but  because  the  line  AB  is  bisected  in  D 
AB  :  DB  :  :  1  :  2, 
g-E  :  gA  :  :  1  :  2, 
AE  :  gA  :  :  3  :  2, 
therefore 

g-A=|AE. 

In  a  triangular  pyramid  the  centre  of  parallel  forces  is  in  the 
line  that  joins  the  vertex  to  the  centre  of  parallel  forces  of  the  base, 
at  the  distance  of  three  fourths  of  that  line  from  the  vertex. 


OP   EQUILIBRIUM.  {Book  L 

Let  ABCD  be  a  triangular  pyramid,  to  the  point  H  that  bisects 

.A. 


the  line  DC,  draw  AH,  BH;  make  HE=i  AH,  HF=iBH; 

E  and  F  will  be  respectively  the  centres  of  parallel  forces  of  the 
surfaces  ADC,  DBC  ;  join  EF,  AF,  BE.  If  the  pyramid  be  re- 
solved into  elements,  by  means  of  planes  parallel  to  DCB,  it  is 
manifest  that  the  line  AF  must  pass  through  the  centres  of  paral- 
lel forces  of  all  the  elements ;  their  common  centre  of  parallel 
forces  must  therefore  be  in  that  line  ;  for  the  same  reason  it  is  the 
line  BF|:  it  is  therefore  at  their  common  intersection,  in  the  point  g. 
By  similar  triangles 

Ag  :  gF  :  :  AB  :  FE  :  :  AH  :  EH, 
AE  :  EH  :  :  3  :  1, 
AH  :  EH  :  :  4  :  1, 

Ag  :  gF  :  :  3  :  1, 

Ag  :  AF: :3:4; 
whence  Ag-=f  AF. 

The  first  of  these  propositions  may  be  applied  geometrically  to 
find  the  centre  of  parallel  forces  of  any  polygon  whatsoever ;  for 
it  may  be  divided  into  a  number  of  triangles,  equal  to  the  number 
of  sides  less  two.  Let  the  centres  of  parallel  forces  be  found  in 
two  of  the  triangles,  join  them  by  a  straight  line,  and  divide  it  into 
parts  inversely  proportioned  to  the  areas  of  the  two  triangles,  the 
point  of  division  is  evidently  their  common  centre  of  parallel 
forces.  If  this  point  be  joined  to  the  centre  of  parallel  forces  of 
the  third  triangle,  and  divided  in  a  similar  manner,  the  point  of  di- 
vision becomes  the  common  centre  of  parallel  forces  of  the  three 
triangles.  This  method  being  continued,  until  all  the  triangles 
into  which  the  polygon  has  been  divided  have  been  used,  the  point 
last  obtained  is  the  centre  of  parallel  forces  of  the  whole  surface. 
A  solid  bounded  by  plane  surfaces  may  be  divided  by  planes, 
into  a  number  of  triangular  pyramids.  The  second  of  these  pro- 
positions therefore  gives  us  a  geometric  method,  founded  on  the 


Book  /.] 


OP    EQUILIBRIUM, 


same  principles,  of  finding  the  centre  of  parallel  forces  in  any  so- 
lid whose  boundaries  are  plane  surfaces. 

27.  When  the  system  of  points  on  which  a  set  of  forces,  whe- 
ther parallel  or  not,  acts,  are  not  so  connected  as  to  form  a  rigid 
and  invariable  system,  the  conditions  of  equilibrium  are  differ- 
ent.    They  may  be  investigated  in  a  general  manner,  but  the 
applications  being  limited  in  practice  to  a  few  cases,  and  the  in- 
vestigation difficult,  we  confine  ourselves  to  the  cases  :  of  poly- 
gons formed  of  invariable  lines,  whose  angles  are  capable  of 
changing  their  magnitudes  under  the  action  of  the  forces  that  are 
applied,  until  the  system  reaches  a  state  of  equilibrium,  and  when 
the  forces  are  parallel  among  themselves  ;  and  of  curves,  formed 
by  the  action  of  an  infinite  number  of  forces,  upon  the  points  of  a 
flexible  but  inextensible  line. 

28.  When  a  system  of  parallel  forces  acts  upon  a  system  of  in- 
flexible and   inextensible  lines,  forced  to  move  around  their  con- 
necting points,  the  sum  of  the  forces  will  be  equal  to  0  ;  the  forces 
will  be  all  in  one  plane,  in  which  the  lines  at  whose  angles  they 
act  are  situated  ;  and  the  forces  will  be  to  each  other  respectively, 
as  the  sum  of  the  cotangents  of  the  angles  their  directions  make 
with  the  lines,  at  whose  point  of  concourse  they  themselves  act. 

Let  a  force  F  act  at  the  angle  A  made  by  two  inflexible  lines 


AT,  AT',  which  angle  is  variable  under  the  action  of  the  forces  ; 
the  system  will  be  in  equilibrio,  when  the  lines  are  drawn,  each  in 
its  respective  direction,  by  forces  T  and  T'  that  are  such  as 
would  be  in  equilibrio  with  F  when  acting  at  the  point  A.  This 
is  evident,  because  we  can  conceive  the  point  of  application  of  a 
force  removed  to  any  point  in  its  direction,  without  changing  its 
action,  provided  the  points  be,  as  in  the  case  before  us,  connected 
by  inflexible  and  inextensible  lines. 

Let  a  and  a'  be  the  angles  the  direction  of  the  force  F  makes 
with  T  and  T';  their  sum  is  the  angle  these  two  forces  make  with 
each  other,  hence  by  (16) 


30 


OF    EQUILIBRIUM. 


[Book  I. 


F  :  T  :  T'  :  :  sin.  a+a' :  sin.  a!  :  sin.  a. 


(39) 


As  three  forces  in  equilibrio  around  a  point  must  be  in  one  plane, 
the  direction  of  the  lines  AT,  AT',  will  be  in  the  same  plane  with 
the  direction  of  F. 

It  will  be  obvious  that  the  more  obtuse  the  angle,  made  by  the 
directions  of  the  lines  becomes,  the  greater  will  be  the  tension  they 
have  to  support  under  the  action  of  the  force  F.  If  the  direction 
of  the  force  F  bisect  the  angle  made  by  the  two  lines,  each  of  them 
will  bear  an  equal  tension. 

Let  us  now  suppose  that,  as  in  the  figure  beneath,  a  number  of 


parallel  forces,  F,  F',  F",  F"',  act  upon  a  system  of  rigid  and  in- 
flexible lines  in  such  a  manner  as  to  cause  equilibrium  ;  resolve 
the  force  F  into  two  others,  T  and  X,  in  the  direction  of  the  two 
lines  at  whose  angles  it  acts  ;  the  second  force  into  two  others,  X' 
and  Y,  in  the  direction  of  the  two  lines  at  whose  angle  it  acts  ; 
resolve  in  like  manner,  F"  into  two  Y'  and  Z  ;  and  F'"  into  Z'  and 
T'.  These  forces  will  have  from  (16)  the  following  relations : 

F  :  T  :  X  :  :  sin.  («+«')  :  sin.  a'  :  sin.  a       "1 
F' :  X' :  Y  :  :  sin.  (6+6')  :  sin.  6'  :  sin.  b       I          .  ,m 
F"  :  Y'  :  Z  :  :  sin.  (c+c')  :  sin.  c' :  sin.  c       ( 
F'"  :Z':T'::  sin.  (d+d')  :  sin.  d'  :  sin.  d  J   . 
But  in  order  that  the  polygon,  formed  by  the  lines  at  whose  angles 
the  forces  act,  shall  be  in  equilibrio,  the  forces  by  which  each  of 
the  lines  is  drawn  at  its  opposite  ends,  must  also  be  in  equilibrio, 
or 

X=X',    Y=Y',    Z=Z'. 

The  several  values  of  these  forces,  obtained  from  the  analogies, 
are  therefore  equal  by  pairs,  or 

F  sin.  a         F'  sin.  6' 


sin.  (aa') 
F'  sin.  6 

sin.  (6-RO: 
F'"sin.  c 

sin.    c+c 


"sin.  (6+6') 
F"  sin,  c' 

=sin.  (c-K)' 
F  sin,  d' 

:sin.  (d+df) 


(41) 


Book  /.]  OF   EQUILIBRIUM.  31 

but  when  the  forces  are  parallel  sin.  a'=  sin.  6,  sin.  6'=  sin.  c, 
sin.  c'  —  sin.  d.  c 

Multiplying  the  equations  just  given  (41),  by  one  or  other  o 
these  equals,  we  have 

F  sin.  a .  sin.  «'__F'  sin.  b  sin.  6' 
sin.  (a+o')"~        sin.  (6+6')"       .  , 

F"  sin,  c  sin.  c'_F'"  sin,  d  sin,    '  ' 
:    sin.  (c+c')  sin.  (d-\-d'} 

but 

sin.  a  sin.  a'     1 

sin.  («+«')  ~~~cot.  a+cot.  a'  ; 

and  so  of  the  rest.  The  expressions  may  therefore  take  another 
form,  and  become 

F  =          F'  ] 

cot.  a-j-cot.  a'~~cot.  6+ cot.  6'       I 
F"  F"'  [ 

~cot.  c+cot.  c'~cot.  d-|-cot.  d'  J    ; 

hence  each  of  the  forces  is  proportioned^  to  the  sum  of  the  cotan- 
gents of  the  two  angles  its  direction  makes  with  the  two  lines,  at 
whose  junction  it  acts. 

The  lines  are  also  in  one  plane,  in  which  the  forces  F,  F',  F", 
&c.  likewise  act. 

The  forces  F,  T  and  X,  being  in  equilibrio  around  a  point,  are 
in  the  same  plane  ;  in  which  X'  lies  also  ;  and  F'  being  parallel  to 
F,  and  drawn  from  a  point  in  the  direction  of  X  is  also  in  the 
same  plane  ;  Y  lies  in  this  plane  also,  because  it  is  in  equilibrio 
with  X'  and  F' ;  and  for  the  same  reason  that  F'  was  in  the  same 
plane  with  F,  F"  lies  in  the  same  plane  with  F' ;  thus  all  the  forces 
lie  in  a  single  plane,  and  the  polygon  is  a  plane  figure. 

The  forces  which  act  to  keep  the  system  in  equilibrio,  being 
parallel,  their  sum  is  equal  to  0  ;  and  if  the  system  be  attached 
at  the  two  extremities,  to  two  fixed  points,  on  which  the  tensions 
resolved  into  two  parallel  directions,  are  T  and  T', 

F+F'+F"+&c.+T+"T/=0. 

The'sum  of  the  parts  of  the  tensions  which  act  in  directions,  paral- 
lel to  those  of  the  forces,  will  therefore  be  equal  to  the  resultant 
of  all  the  other  forces. 

29.  When  the  polygon  has  its  two  extremities  fixed,  it  is  called 
the  Funicular  Polygon,  because  it  is,  as  will  be  hereafter  shown, 
the  figure  a  rope  would  assume  when  loaded  by  weights  attached 
to  different  points  of  its  length.  When  the  points  at  which  the 
forces  act  become  infinitely  near,  the  polygon  becomes  a  curve 
that  may  be  called  the  funicular  curve  ;  and  when  the  weights  are 
equal,  the  curve  is  called  the  Catenaria.  The  research  of  the  equa- 
tions of  the  Catenaria  is  not  strictly  an  object  of  elementary  me- 


34  or  EQUILIBRIUM.  [Book  I. 

R=V(Xa+Y2); 

call  the  angles  its  direction  makes  with  the  two  axes,  at  the  point, 
in  which  if  produced,  they  will  meet,  a  and  /3.  Then  by  (18), 

X          a     Y 

cos.  a=—  ,cos./3=—  • 

All  that  remains  is  to  determine  the  co-ordinates  of  the  point 
of  application.  These  may  be  determined,  when  the  position 
of  the  points  of  application  of  the  several  forces  are  determined, 
and  their  co-ordinates  given,  by  means  of  the  principle 
contained  in  formulae  (34,  and  35).  In  these,  the  values  of 
the  several  forces,  with  those  of  their  co-ordinates,  are  to  be  sub- 
stituted in  a  manner  too  obvious  to  heed  description. 

:'M»OflOO 

31.  The  value  of  the  resultant  .of  forces  acting  in  one  plane, 
may  also  be  determined  by  means  of  what  are  called  their  Mo- 
ments of  Rotalion. 

To  understand  the  meaning  of  this  term,  we  shall  recur  to  the 
investigation  of  the  value  of  the  resultant  of  two  parallel  forces, 
(§  22.)  In  the  course  of  that,  it  was  found  that  the  perpendicu- 
lar distances  of  the  directions  of  two  converging  forces,  acting 
upon  an  inflexible  line,  from  the  point  of  application  of  their  result- 
ants, and  the  forces  themselves,  were  in  inverse  proportion,  or  as 
represented  by,  (25), 

A  :  B  :  :  b  :  a, 
hence, 


With  two  forces,  having  this  relation,  a  third  applied  to  the  point  of 
application  of  the  resultant,  equal  to  it,  and  opposite  in  direction, 
would  cause  the  system  to  be  in  cquilihrio.  Let  us  now  suppose 
that  instead  of  applying  a  force  to  this  point,  it  becomes  fixed,  but 
in  such  a  manner  that  the  line  of  application  of  A  and  B,  may  be 
free  to  revolve  around  it  as  a  centre  of  motion.  The  two  forces 
although  unequal  in  magnitude,  are  still  in  equilibrio,  and  each 
will  tend  to  make  the  line  revolve  with  equal  energy.  This  energy, 
then,  may  be  expressed  by  the  two  equal  products  A«,  and  B6  ; 
and  in  general,  if  we  suppose  the  point  of  application  of  any  force 
to  be  united  to  a  fixed  point,  by  an  inflexible  line,  the  force  will 
act  to  cause  the  line  to  revolve  around  that  poinf,  with  an  energy 
determined  by  the  product  of  its  intensity,  into  the  perpendicular 
distance  of  the  fixed  point,  from  the  direction  of  the  force  ; 
hence  : 

32.  The  moment  of  rotation  of  a  force,  in  respect  to  a  point, 
is  the  product  of  the  intensity  of  the  force  into  the  perpendicu- 
lar let  fall  upon  the  direction  of  the  force. 

33.  The  moment  of  rotation  of  two  forces,  in  respect  to  any 
point  situated  in  their  plane,  is  equal  to  the  sum  or  difference  of 


/.] 


OP    EQUILIBRIUM. 


35 


the  moments  of  rotation  of  its  components,  in  respect  to  the  same 
point;  to  the  difference,  when  the  point  falls  within  the  angle, 
formed  by  the  directions  of  the  two  components  ;  to  the  sum 
when  the  point  falls  without  this  angle. 

Let  F  and  F'  be  the  two  forces,  R  their  resultant,  converging 
to  the  point  d  ;  C  the  point  whence  the  perpendiculars  are  let  fall 


let/,/,  and  r,  be  the  three  perpendiculars  ;  let  the  distance  C  d, 
=c.  Let  each  of  the  forces  F,  F',  and  R',  be  decomposed  in  two 
others,  one  in  the  direction  of  c,  the  other  perpendicular  to  it, 
or  in  the  direction  B  d  E. 

The  value  of  the  component  of  R,  in  the  direction  BdE, 
will  be 

Rcos.  ^RcZB; 
but 

•  'iO  o: 
,    cos. 


and  the  component  of  R,  in  the  direction  of  B  d  E,  becomes 

:-;fcem|  '• 


In  the  same  manner,  the  components  of  F  and  F',  in  the  same 
direction  B  d  E,  may  be  shown  to  be 


, 


these  will  be  in  the  same  direction,  when  the  point  C  falls  without 
the  angle,  and  in  opposite  directions  when  it  falls  within.  Now,  aa 
these  three  forces  would  be  in  equilibrio,  if  R  were  applied  in  a 
reversed  direction ;  their  components  in  relation  to  two  rectan- 


36  OP    EQUILIBRIUM.  [Book  I. 

gular  axes,  would  be  equal  also  ;  the  components  of  F  and  F',  are 
therefore  equal  to  the  component  of  R,  and 

R.:=F/+F/; 
c         c*        c 

multiplying  by  c 

R,=F/+F/, 

which  expresses  our  proposition. 

Extending  the  investigation  in  the  usual  manner  to  any  num- 
ber of  forces,  we  have 


,  (45) 

In  this  expression,  it  is  obvious  that  the  signs  +  and  —  ,  express 

the  tendency  of  the  force  to  turn  the  system,  in  one  or  the  other 

direction,  around  the  centre  of  the  Moments. 
In  case  of  equilibrium,  the  expression  becomes 

F/+F'/+F"/'  HF&c.  -0,  (46) 

or  : 

34.  A  system  of  forces,  acting;  in  one  plane  upon  a  system  of 
points  invariably  connected,  will  be  in  equilibrio,  when  the  sum  of 
the  moments  of  all  the  forces  that  tend  to  make  the  system  turn  in 
one  direction,  is  equal  to  the  sum  of  the  moments  of  all  the  forces 
that  tend  to  make  the  system  turn  in  an  opposite  direction.  The 
same  proposition  is  also  true  in  the  case  of  the  system  being 
firmly  attached  to  the  point  C,  or  to  the  centre  of  the  moments  ;  for 
were  the  two  sets  of  moments  unequal,  one  or  the  other  would 
preponderate,  and  would  make  the  system  revolve.  In  this  case 
it  is  not  necessary  that  the  resultant  of  the  forces  should  be  equal 
to  0,  but  merely,  that,  as  the  moment  of  the  resultant  is  equal  to  the 
sum  or  difference  of  the  moments  of  all  the  forces, 

Rr=0. 
But  this  can  only  happen,  if  R  have  any  magnitude,  when 

r=0; 
hence  the  resultant  must  pass  through  the  fixed  point;  and, 

35.  When  a  system  of  forces  is  applied  to  a  system  of  points 
invariably  connected  together  in  one  plane,  and  having  one  fixed 
point,  the  direction  of  their  resultant  must  pass  through  that  point, 
or  the  system  will  not  be  in  equilibrio. 

36.  If  we  suppose  a  straight  line  to  be  drawn  through  the 
centre  of  the  moments,  and  perpendicular  to  the  plane,  this  line 
will  become  an  axis,  on  which  the  forces  would  tend  to  make  the 
system  revolve  ;  and  it  will  be  no  longer  necessary  that  the  forces 
should  act  in  one  plane,  but  merely  that  they  act  in  planes  parallel 
to  each  other  ;  for  their  moments,  determined  by  lines  drawn  per- 


/.]  OP    EQUILIBRIUM.  37 

pendicular  both  to  their  own  direction  and  the  axis,  would  remain 
constant. 

37.  If  the  forces  do  not  act  in  planes  perpendicular  to  the  fixed 
axis,  each  of  them  may  be  resolved  into  two,  one  parallel  to  the 
axis,  the  other  lying  in  a  plane  perpendicular  to  it.      It  will  then 
be  obvious,  that  the  former  produces  no  effect  to  make  the  system 
revolve ;  and  no  more  of  the  force  is  exerted,  for  that  purpose, 
than  is  represented  by  the  latter.      The  line  that  represents  that 
component  of  a  given  force,  which  acts  in  a  plane  different  from 
that  in  which  it  is  itself  situated,  corresponds  with  its  geometric 
projection  in  that  plane  ;  and  calling  the  force  A  ;  the  projection 
P ;  and  the  angle  the  two  planes  make  with  each  other  i ;  this 
force  may  be  found  (14)  by  the  formula. 

P=A  cos.  i.  (47) 

38.  The  conditions  of  equilibrium  in  forces  acting  upon  points 
invariably  connected,  also  hold  good,  when  the  points  are  con- 
nected in  any  manner  whatsoever;  for  it  is  evident  that  if  the 
system  be  in  equilibrio,    the  state  of  equilibrium  will  not  be 
changed,  by  uniting  the  points  of  which  it  is  composed  in  an  inva- 
riable manner.     But  in  addition  to  the  conditions,  that  are  alone 
necessary  in  points  connected  in  an  invariable  manner,  and  are 
common  to  systems  connected  in  any  manner  whatsoever,  there 
will  be  others,  that  will  depend  upon  the  manner  in  which  the 
points  are  connected. 


•>fi<te  ciaad  91  < 


BOOH.  II. 

OF   MOTION. 

ta 

CHAPTER  I. 

Op    MOTION    IN    GENERAL.       UNIFORM    MOTION.       GENERAL 

PRINCIPLES  OP  VARIABLE  MOTION. 

39.  When  a  material  point  is  acted  upon  by  forces  under  whose 
action  it  is  not  in  equilibrio,  it  is  set  in  motion,  as  it  also  would 
be  by  the  action  of  a  single  force.  If  it  be  not  acted  upon  by  any 
force,  as  there  is  no  reason  that  it  should  move  in  one  direction 
rather  than  another,  it  will  remain  at  rest ;  so  also  when  once  set 
in  motion,  and  no  force  act,  or  if  the  forces  that  do  act  are  in 
equilibrio,  it  must  continue  to  move  uniformly  forwards  in  a 
straight  line  ;  for  there  is  now  no  reason  why  it  should  change 
either  the  rate,  or  the  direction  of  its  motion.  Hence,  all  bodies 
will  continue  in  the  same  state,  either  of  rest,  or  of  motion  uni- 
formly forwards  in  a  straight  line;  unless  they  be  compelled  to 
change  their  state  of  rest  or  of  motion,  by  the  action  of  some  force 
impressed  upon  them.  The  truth  of  this  principle  is  not  obtained 
from  abstract  reasoning  alone,  but  is  the  uniform  result  of  obser- 
vation and  experience.  Although  from  what  is  observed  to  occur 
in  moving  bodies  near  the  surface  of  the  earth,  we  might  at  first 
sight  infer  that  they  had  a  natural  tendency  to  come  to  rest ;  still, 
when  we  remark,  that  the  more  we  lessen  the  resistances,  the 
longer  is  the  continuance  of  the  motion;  and  that  we  can  in  al- 
most all  cases,  ascribe  the  diminution  of  the  motion,  or  its  change 
of  direction,  to  forces  that  we  know  from  other  circumstances  to 
be  acting;  we  infer,  that  were  these  resistances  not  to  act,  the 
body  would  go  on  uniformly  in  a  straight  line.  This  principle 
is  sometimes  ranked  as  a  property  of  matter,  and  is  called  its 
Inertia.  A  similar  inference  may  be  deduced  from  the  motions 
of  the  heavenly  bodies.  In  these,  since  the  earliest  record  of  au- 
thentic observation,  no  change  has  been  detected,  that  is  not  pe- 


40  OF    MOTION.  [Book  II. 

riodic  ;  their  mean  rates  of  motion  have  therefore  been  constant, 
although  no  force  has  been  applied  to  maintain  their  motions. 

40.  In  order  to  represent  the  circumstances  of  motion,  which 
consist  in  the  passage  of  a  body  through  a  portion  of  space  in  some 
definite  time,  we  make  use  of  the  term  Velocity.  Velocity  is  the 
relation,  or  ratio,  of  the  spaces  described  to  the  times  employed 
in  describing  them.      It  Will  be  obvious  (hat  in  different  motions, 
such  a  relation  does  actually  exist ;  the  more  rapid  the  motion, 
the  less  being  the  time  occupied  in  describing  a  given  space,  and 
the  greater  the  space  described  in  a  given  time.      But  space  and 
time  are  essentially  heterogeneous  quantities,  and    incapable  of 
any  direct  comparison.    We  are  therefore  compelled  to  resort  to  a 
means  of  comparison,  by  adopting  a  method,   in  which  the  mea- 
sure of  each  of  these  quantities  may  be  considered  as  an  abstract 
number.      Between  such  numbers,   a  ratio  capable  of  being  ex- 
pressed does  exist.      In   order  to  effect  this  object,  we  assume 
some  conventional  unit  for  the  space,  as  the  foot,  for  instance  :  in 
like  manner  we  assume  a  conventional  unit,  for  the  measure  of  the 
time.      In  terms  of  these,  a  ratio,   or  relation   may  be  expressed. 
As  the  velocity,  in  uniform  motion,  increases  with  the  space  de- 
scribed, while  the  time  diminishes,   the  relation  between  them 
may  be  thus  expressed  : 

.=4      '  (48) 

where  v  is  the  velocity,  s  the  space,  and  /  the  time.  The  first 
of  these  will  be  denoted  in  the  number  of  the  conventional  units 
of  space,  described  by  the  body  in  the  unit  of  time. 

From  this  equation  we  obtain  expressions  for  the  value  of  the 

space  s,  and  the  time  /.  as  follows  : 

k  8 

s=v  /,          f=-.  (49) 

oil 

41.  When  two  uniform  motions  are  to-be  compared  together  ; 
as  for  instance,  when  we  wish   to  ascertain   the  time   in   which 
bodies  moving,  or  appearing  to  move,  in  the  same  line  with  dif- 
ferent velocities,  shall  be  at  the  same  point,  or  shall  appear  to  meet; 
we  may  estimate  the  spaces,  from  some  given  point  in  the  line, 
actually,  or  apparently  described,  by  the    two    bodies.     Let  s  re- 
present the  distance  from  the  fixed  point  at  the  time  /  ;  b  the  dis- 
tance from  the  same  point,  at  the  instant  from  which  the  time  /  is 
estimated  ;   then  the  space  becomes  s — b  j  and  the  first  equation 
(48)  becoir.es  ,)n'l 


Book  //.]  OF  MOTION.  41 

whence, 

s=vt+b.  (51) 

The  time  /,  may  be  either  positive  or  negative  ;  when  positive,  it 
denotes  intervals  of  time  subsequent  to  the  passage  of  the  body 
through  the  fixed  point  ;  when  negative,  it  denotes  intervals  prior 
to  the  body's  reaching  that  point.  So  also,  may  s  have  positive 
or  negative  values,  which  represent  its  position  in  respect  to  the 
fixed  point.  In  this  manner  the  equation  (51)  will  point  out  the 
position  of  the  body  for  every  possible  instant  of  time,  in  the  line 
that  marks  out  the  direction  of  its  motion. 

If  there  be  another  body  moving,  or  appearing  to  move,  uni- 
formly in  the  same  line  with  the  first,  whose  velocity  is  v'  ;  whose 
distance  at  the  same  time  denoted  by  t,  from  the  fixed  point  is  s'  ; 
and  the  distance  from  the  same  point  at  the  instant  whence  /  is 
estimated,  is  b',  the  equation  of  its  motion  will  be 

s'=v't+b.'  (52) 

In  this  equation,  besides  the  same  relations  of  positive  and  ne- 
gative, among  the  quantities,  that  we  have  pointed  out  as  appli- 
cable to  the  former,  the  quantity  v'  will  be  negative,  when  the  mo- 
tion is  in  a  direction  contrary  to  that  of  the  first  body. 

A  comparison  between  the  values  of  these  two  equations,  will 
point  out  the  relative  position  of  the  two  bodies  in  the  direction 
of  their  motion.  When  both  are  at  the  same  instant,  at  the  same 
distance  from  the  fixed  point,  s=s',  whence, 


which  gives  for  the  value  of  / 

6'—  6 


If  this  value  should  be  found  negative,  it  denotes,  that  the  bodies 
meet  before  the  instant  whence  the  time  is  computed. 

To  give  an  instance  of  the  application  of  these  formulae  :  sup- 
pose  two  bodies  to  be  moving  in  the  same  line,  and  in  the  same 
direction,  with  velocities  in  the  relation  of  1000  :  1. 

Then,  v  =  1000,  «'  =  1. 

Let  A,  represent  the  body  whose  velocity  is  greatest  ;  B,  that 
whose  velocity  is  least. 

Let  A  be  situated,  at  the  instant  whence  the  time  is  estimated, 
at  the  fixed  point  ;  then, 

6=0; 
let  B  be  situated  at  the  same  instant,  at  a  distance  represented  by 

6'  =  1000, 
then, 

V—  6=1000 
v  —  1/=  999 

6 

»  < 


42  or  MOTION.  [Book  11' 

b'—b      1000         1 

99=1999  =  1-001' 

vb'  _  1000000 
—  _—  _ 

=  1001555  =  1001,001; 

hence  the  body  A,  will  overtake  the  body  B,  at  the  end  of  one  of 
the  units  of  time,  and  the  e^th  part  of  that  unit  ;  and  will  have 
described  a  space  equal  to  1001  g±-$  units  of  space.  But,  esti- 
mated decimally  1  the  fractions  become  the  infinite  converging 
series  .  00  1  00  1  .  The  ancients,  unaware  of  the  fact,  that  the  sum 
of  such  an  infinite  series,  was  a  finite  quantity,  reasoned  from  it, 
to  show  that  the  one  body  would  never  overtake  the  other.  The 
argument  employed  by  them,  was  called  the  Achilles,  and  the 
conditions  were  the  same  as  those  we  have  chosen  for  our 
example. 

The  same  propositions  may  be  applied  to  uniform  motions,  in 
any  lines  whatsoever  ;  as  for  instance  to  motion  in  re-entering 
curves,  of  which  the  circle  would  furnish  the  most  simple  instance  ; 
and  the  hour,  minute,  and  second  hand  of  a  watch,  supposed  to 
be  fixed  upon  the  same  arbor,  would  afford  an  apt  illustration. 

Supposing  them  all  to  set  off  from  the  same  point,  6  —  6'  and 
6'  —  b"  become  equal  to  the  circumference  of  the  circle  or  *r,  and 
for  two  hands, 


for  all  three  hands 


the  application  of  this  to  calculation,  is  too  obvious  to  require  an 
example. 

42.  If  a  point  be  impressed  at  the  same  time  with  two  uni- 
form motions,  it  moves  in  a  line  which  is  determined  by  their 
joint  effect ;  and  as  each  of  these  motions  is  due  to  a  force,  the 
point  will  move  as  if  it  were  actuated  by  a  single  force,  which  is 
the  resultant  of  the  two  forces  ;  hence,  from  the  principles  of  §  12, 
it  must  move  in  the  diagonal  of  the  parallelogram,  constructed 
upon  the  two  forces  as  sides.  If  actuated  by  any  number  of  motions 
whatsoever,  it  will  move  in  the  direction  of  the  resultant  of  the 
forces  that  cause  these  several  motions. 

Instances  of  bodies  that  are  actuated  at  the  same  time,  by  more 
than  one  motion,  are  innumerable  in  practice.  All  bodies  re- 
tained upon  the  surface  of  others,  by  means  of  attraction,  friction, 
or  any  other  force,  acquire  the  motion  of  the  bodies  on  which 
they  rest. 


//.]  OF    MOTION.  43 

Thus  a  body  in  a  carriage,  or  mounted  upon  a  horse,  a  person  in 
a  vessel,  and  finally  all  bodies  upon  the  surface  of  the  earth,  have 
a  motion  due  to  that  of  the  body  on  which  they  rest.  This  be- 
comes evident  upon  a  sudden  cessation  of  the  motion,  and  is  not 
instantly  communicated,  as  may  be  perceived  immediately  after 
the  beginning  of  the  motion.  Thus,  when  a  steam-boat  is  sud- 
denly set  in  motion,  we  feel  a  tendency  to  move  in  a  direction 
apparently  opposite ;  this  is  due  to  the  inertia,  which  would 
leave  us  in  our  original  position  in  respect  to  fixed  objects  on  the 
shore,  were  not  the  motion  of  the  vessel  communicated  to  us  ;  and 
when  the  same  vessel  has  her  progress  suddenly  checked,  we  ex- 
perience a  tendency  to  move  forwards,  that  remains  until  again 
counteracted,  in  the  mode  in  which  the  motion  was  first  com- 
municated. 

A  body  thrown  from  a  carriage  In  rapid  motion,  by  a  force  act- 
ins;  perpendicular  to  the  direction  of  the  motion,  does  not  fall  to 
the  ground  opposite  to  the  point  where  the  carriage  was,  when  it 
was  projected,  but  opposite  to  the  point  the  carriage  has  reached, 
at  the  time  it  falls  to  the  ground. 

In  feats  of  horsemanship,  balls  are  thrown  up  vertically,  at  least 
so  far  as  the  action  of  the  rider  influences  them,  but  being  im- 
pressed by  the  motion  of  the  horse,  they  fall  again  into  the  hand 
of  the  rider  ;  the  rider  may  spring  directly  upwards  from  the 
saddle,  and  fall  again  upon  it,  although  the  horse  be  at  full  speed. 
The  directions  in"  which  bodies  move,  being  influenced  by  all  the 
motions  with  which  they  are  impressed,  the  position  of  bodies 
moving  upon  surfaces  that  are  themselves  in  motion,  is  the  same 
in  relation  to  points  in  the  moving  surface,  at  given  instants  of 
time,  as  it  would  have  been,  had  the  surface  remained  at  rest. 
Thus  we  move  from  place  to  place,  upon  the  deck  of  a  vessel, 
whose  motion  is  not  disturbed,  with  precisely  the  same  effort, 
that  we  would  perform  the  same  distance  upon  the  land  ;  and  were 
we  not  aware  of  the  vessel's  motion  from  other  circumstances, 
would,  as  is  done  by  children,  ascribe  the  motion  to  the  surround- 
ing objects. 

The  same  happens  to  us,  from  our  situation  on  the  surface  of 
the  earth.  This  is  impressed  with  a  rapid  motion,  not  only  of 
revolution,  but  of  translation  ;  yet  these  being  to  all  intents  uni- 
form, we  ascribe  the  change  of  apparent  position  that  our  motion 
causes  in  the  heavenly  bodies,  to  a  proper  motion  existing  in  them  ; 
and  it  was  ages  after  these  apparent  motions  had  been  carefully  ob- 
served by  astronomers,  before  the  true  cause  of  the  phenomena 
was  detected. 

43.  Although  it  is  the  tendency  of  matter,  if  once  set  in  motion, 
to  move  forwards  forever  with  uniform  velocity,  in  the  same  direc- 


44  OP  MOTION.  [Book  II. 

tion,  this  is  by  no  means  the  most  frequent  case  of  motion  that 
occurs  in  nature.     There  are  in  fact  two  distinct  species  offerees. 

(1)  Those  which  having  acted  for  a  time  upon  a  body,  abandon  it 
and  leave  it  to  go  forward,  so  far  as  they  are  concerned,  in  a  right 
line  with  uniform  velocity;  these  are  called  projectile  forces. 

(2)  Those  which  act  during  the  whole  continuance  of  the  body's 
motion.     Such  forces  will  cause  changes  in  the  direction  and  ve- 
locity of  the  body,  and  produce  what  in  general  terms,  are  called 
Variable  Motions. 

44.  As,  when  a  point  that  has  at  one  instant  of  time  been  at  rest, 
is  afterwards  found  in  motion,  we  infer  the  action  of  some  force 
to  cause  that  motion  ;  so,  when  a  point  in  motion,  whose  velocity 
and  direction  have  been  determined,  at  some  instant  of  time,  is 
afterwards  found  moving  in  a  different  direction,  or  with  differ- 
ent velocity,  we  also  infer  the  action  of  some  force  to  produce 
this  change.     This  force  may  either  have  acted  for  a  greater  or 
less  time,  and  then  abandoned  the  point  to  itself,  or  it  may  have 
been  continually  acting.   A  force  of  the  latter  description  is  called 
an  Accelerating  Force,  because  had  it  acted  upon  a  point  origi- 
nally at   rest,   it  would   have   given   the    point   an   accelerated 
velocity. 

We  judge  of  the  fact  of  the  motion  of  a  point  being  accelerated, 
by  comparing  the  spaces  described  in  equal  times  ;  if  after  having 
described  a  certain  space  in  the  unit  of  time,  it  shall  be  found 
describing  a  greater  space  in  an  equal  time,  its  motion  has  evi- 
dently been  accelerated.  So,  when  after  having  described  a  given 
space  in  the  unit  of  time,  it  is  afterwards  found  describing  a  less 
space  in  the  same  time,  its  motion  is  retarded.  An  accelerating 
force  may  produce  a  retarded  motion  ;  for  it  may  act  in  a  direction 
contrary  to  the  motion  originally  impressed  by  some  other  force 
upon  the  body  ;  and  indeed  most  retarded  motions  are  due  to  the 
action  of  forces  that  may  be  considered  under  the  general  head  of 
Accelerating. 

45.  The  time  which  a  point  takes  to  describe  a  given  space, 
will  obviously  depend  upon  the  intensity  of  the  force  that  causes 
its  motion.  Thus  a  force  of  double  the  intensity,  will  cause  a  point 
to  describe  twice  the  space  in  an  equal  time,  and  so  on.     As  we 
know  absolutely  nothing  of  the  nature  of  forces,  but  find  their 
action  to  be  proportioned  to  the  spaces  described,  we  might  take 
the  spaces  described  in  equal  times  as  the  measure  of  the  forces  ; 
but  as  the  velocities  are  proportioned  to  the  spaces,  they  are  also 
proportioned  to  the  forces  ;  and  therefore  in  uniform  motions, 
where  mere  material  points  are  concerned,  the  velocity  and  force 
may  mutually  serve  as  each  other's  measures.     All  the  princi- 
ples and  formulae  that  have  been  applied  in  the  previous  book  to 


Booh  //.]  OP  MOTION.  45 

the  composition  offerees,  are,  therefore,  applicable  to  the  compo- 
sition of  velocities. 

46.  In  motions  produced  by  a  force  that  acts  continually,  the 
velocities  are,  as  we  have  seen,  variable.  It  is  impossible  for  us 
to  determine  from  experience  alone,  whether  such  a  force  acts  with- 
out interruption,  or  whether  it  produces  its  effect  by  a  succession  of 
impulses,  separated  by  inappreciable  intervals  of  time.  Which- 
ever of  these  modes  of  action  be  the  true  one,  the  results  in  both 
cases  will  be  the  same  ;  for  if  we  suppose  the  velocities  to  be  re- 
presented by  the  ordinates  of  a  curve,  whose  abscissas  represent 
the  times;  the  motion  produced  by  a  succession  of  impulses,  sepa- 
rated by  infinitely  small  intervals  of  time,  would  correspond  to  a 
polygon  of  an  infinite  number  of  sides;  and  this  would  be  identical 
with  the  curve.  We  therefore  consider  all  variable  motions,  as 
made  up  of  a  succession  of  uniform  motions,  each  continued  fora 
very  short  space  of  time  ;  and  the  results  obtained,  from  investi- 
gations founded  on  this  principle,  are  indentical  with  those  that 
would  occur,  were  the  velocity  to  be  continually  varying. 

In  the  equation  (48)  the  time  and  space  s  and  tf,  becoming  in- 
finitely  small,  will  be  represented  by  their  differentials,  and 

„=*,*=.*,  *  =£,  (53) 

The  intensity  of  an  accelerating  force,  cannot,  like  that  of  a 
force  that  produces  uniform  motion,  be  measured  by  the  velocity, 
for  that  is  always  varying,  even  although  the  force  may  remain 
constant  ;  but  it  may  be  measured  by  the  momentary  variations 
in  the  velocity.  Let  dv  be  the  increase  of  the  velocity  during  the 
time  dt  ;  then,  since  the  increase  in  the  velocity  will  be  the  same 
as  it  would  have  been,  had  the  action  of  the  accelerating  force  not 
been  interrupted,  dv  will  be  equal  to  the  product  of  the  force  into 
its  time  of  action  dt,  or  calling  the  force/, 

dv=fdt, 
and 


then  since  by  (53) 
we  have 


<«> 


Such  are  the  general  equations  of  variable  motion  ;  and  they 
are  applicable  either  to  the  case  of  its  being  accelerated  or  retarded  ; 
but  in  the  latter  case,  dv  is  a  negative  quantity. 


OF    MOTION.  [Book  II. 


CHAPTER  II. 

OF  RECTILINEAL  MOTION  UNIFORMLY  ACCELERATED,  OR  UNIFORMLT 

RETARDED. 

47.  When  the  spaces  that  a  point  describes  in  equal  times, 
increase  or  decrease  by  equal  increments  or  decrements,  the  mo- 
tion is  said  to  be  uniformly  accelerated  or  retarded.     In  such  a 
case  the  quantities  dv,  and  dt,  in  equation  (54),  are  obviously  con- 
staat;  hence  the  accelerating  force/",  is  a  constant  force. 

48.  We  call  the  velocity  that  a  body  would  have,  were  the  ac- 
celerating force  removed,  its  Final  Velocity  ;  or,  as  we  consider  the 
body  to  move  uniformly,  for  infinitely  small  intervals  of  time,  it 
may  be  considered  at  its  velocity  as  the  end  of  the  given  time, 
and  used  without  the  addition  of  the  word  final.     The  space  de- 
scribed from  rest,  in  acquiring  that  velocity  under  the  action  of 
the  accelerating  force,  iscalled  theSpace  due  to  that  Velocity  ;  and 
the  velocity  acquired,  is  called  the  Velocity  due  to  the  Space. 

49.  In  the  motion  of  a  point  uniformly  accelerated  from  a  state 
of  rest,  the  velocities  are  proportioned  to  the  times;  the  whole 
spaces  described,  are  proportioned  to  the  squares  of  the  times  ; 
and  if  the  times  be  represented  by  the  series  of  natural  numbers, 
the  acquired  velocities  will  be  represented  by  the  series  of  even 
numbers  ;  the  whole  spaces,  by  the  series  of  square  numbers ;  and 
the  spaces  described  in  the  successive  units  of  time,  by  the  series 
of  odd  numbers.     The  measure  of  the  accelerating  force  is  equal 
to  twice  the  space  described  under  its  action  from  rest,  in  the 
first  unit  of  time  ;  and  if  the  accelerating  force  be  removed,  the 
velocity  acquired  in  a  given  time  by  its  action,  is  such  as  would 
carry  the  point  in  an  equal  time,  with  uniform  motion,  through 
twice  the  space  it  has  passed  through,  in  acquiring  that  velocity. 

Suppose  the  point  to  have,  at  the  instant  the  accelerating  force 
begins  to  act,  and  in  the  same  direction,  a  velocity = a.  It  will  ac- 
quire during  each  successive  unit  of  time,  an  additional  velocity, 
which  as  the  force  is,  in  the  case  under  consideration,  a  constant 
one,  will  be  a  constant  increment.  This  increment,  which  will  be 
the  measure  of  the  accelerating  force,  we  shall  call  g.  The  velo- 
city being  originally  a,  will  become,  at  the  end  of  the  first  unit  of 
time,  a+g ; 

at  the  end  of  the  second  unit,  «-f2g-; 

and  at  the  end  of  the  time  /,  a~^~gt 5 

or  calling  the  final  velocity  v, 

v^a+gt.  (57) 


jBook  //.]  OP  MOTION.  47 

Ift  vary,  and  become  t+dt,  the  space  described  »,  will  vary 
also,  and  become  s+ds  :  ds  then,  is  the  space  described  in  the  time 
dt. 

If  we  suppose  that  for  this  small  interval  of  time,  the  velocity  is 
constant,  and  equal  to  v,  we  have  by  (53), 

ds=vdt, 
substituting  the  value  of  v  from  (57) 

ds=adt+gtdt. 
Integrating 


b  being  the  distance  from  some  fixed  point  in  the  direction  of  the 
motion  ;  but  when  no  more  than  a  single  point  is  concerned,  6 
may  be  taken  equal  to  0,  and  the  equation  becomes 

fif 

s=at+~  (58) 

eliminating  t  by  means  of  equation  (57). 
v2—  a2 


If  the  body  be  at  rest  when  the  accelerating  force  begins  to  act, 
a=0,  and  the  equations  (57),  (58),  (59),  become 


from  these  equations  we  readily  obtain  others,  which  with  them, 
give  the  value  of  the  several  quantities,  each  in  terms  of  two  of 
the  others,  as  follows,  viz. 


2s      2s 


lit 


(61) 

- 


If  the  motion  continue  only  for  the  unit  of  time 


v=2s. 


The  expression  v=gt,  in  which  g  is  a  constant  quantity,  shows 
us  that  the  velocities  acquired  are  proportioned  to  the  times. 

gf  g 

The    expression    *=~2~   in  which  ^  is  constant,  shows  u» 


48  or  MOTION.  [Book  It. 

that  the  whole  spaces  are  proportioned  to  the  squares  of  the 
times. 

2s 
Applying  the  formula  «=•£   to  times  taken  as*  the  series  of 

natural  numbers,  and  calling  the  space  described  in  the  first  unit 
of  time,  unity  ;  we  have,  for  the  successive  values  of  t>, 

2,  4,  6,  8,  &c. 
On  the  same  hypothesis  we  have  for  the  values  of  5, 

1,4,9,16,  &c. 

The  successive  differences  of  this  series  represent  the  spaces 
described,  during  each  successive  unit  of  time,  and  are, 

1,  3,  5,  7,  &c. 

The  expression  g=2s,  when  the  motion  has  continued  from 
rest,  for  the  unit  of  time,  shows  that  the  measure  of  the  accele- 
rating force  is  equal  to  twice  the  space  described,  from  rest,  in  the 
first  unit  of  time. 

2s 
And  the  expression  v=~^~  compared  with  the  equation  (48)  of 

s 

uniform  motion,  v= j  shows  us,  that  a  point,  with  the  velocity  ac- 
quired by  moving  from  rest  under  the  action  of  a  constant  force, 
would  pass  through  twice  the  space  in  an  equal  time. 

50.  In  the  motion  of  a  point  that  is  equably  retarded  by  the 
action  of  a  constant  force,  the  velocities  are  as  the  times  that 
remain  until  the  cessation  of  the  motion  ;  and  the  spaces  that 
remain  to  be  described,  are  as  the  squares  of  the  times  estimated 
in  the  same  manner,  and  as  the  squares  of  the  velocities.  The 
point  will  go,  before  it  loses  its  motion,  through  half  the  spacs 
that  it  would  have  described,  in  the  remaining  time,  with  uniform 

velocity. 

«t 

In  our  previous  investigation  if  applied  to  this  case,  g  becomes  a 
negative  quantity ;  for  the  accelerating  force  acts  in  opposition  to 
the  original  velocity.  Hence  in  retarded  motion, 


(61a) 


Book  IL]  OF    MOTION.  48 

when  v=0,  at  which  time  the  constant  force  will  have  completely 
destroyed  the  initial  velocity, 

a  a3  a* 


from  which  equations,  the  principles  we  have  stated  can  be  rea- 
dily determined. 

5t.  If  the  original  direction  of  the  motion  be  not  in  the  same 
straight  line,  in  which  the  accelerating  force  acts,  the  point  must 
describe  a  curve.  For  it  will  in  the  first  instant  of  time  describe 
the  diagonal  of  a  parallelogram,  whose  sides  represent  the  uniform 
velocity,  and  the  measure  of  the  accelerating  force;  in  this  di- 
rection it  would  tend  to  go  forward,  were  it  not  again  acted  upon 
by  the  accelerating  force  ;  this  action  would  produce  a  second 
deflection,  and  so  on  ;  and  thus  the  point  would  describe  a  poly- 
gon of  an  infinite  number  of  sides,  or  a  curve.  Before  then  we 
can  determine  the  nature  of  the  line  the  body  describes,  it  be- 
comes necessary  to  investigate  the  general  properties  of  curvi- 
linear motion. 


IK)  OP  MOTION.  [Book  II. 

CHAPTER  III. 
OF  CURVILINEAR  MOTION. 

52.  The  forces  which  concur  to  produce  curvilinear  motion 
are,  in  general,  resolved  into  three,  parallel  to  three  fixed  rectan- 
gular axes.  This  may  be  done,  as  explained  in  §  16,  in  respect 
to  any  forces  whatsoever.  The  moving  point  will  describe  the 
resultant  of  these  three  forces,  which  will  identically  replace  all 
the  others  that  act.  In  the  small  elements  of  the  time,  it  will 
describe  the  diagonal  of  a  parallelepiped,  of  which  the  three  rec- 
tangular forces,  considered  as  producing  uniform  motion  for  the 
small  interval  of  time,  are  the  sides.  Now  if  we  suppose  perpen- 
diculars to  be  let  fall,  from  the  successive  positions  of  the  moving 
point,  upon  the  three  axes,  the  points  in  which  these  perpendicu- 
lars cut  theaxes,  will  move  along  with  them;  and  the  motion  of  each 
of  them  will  be  such  as  would  be  due  to  a  rectangular  force,  that 
acts  parallel  to  that  axis.  If  any  of  the  forces  cease  to  act,  the 
motion  in  the  directions  parallel  to  the  other  axes  will  not  be 
changed.  Each  of  the  three  rectangular  forces,  into  which  all 
that  act  are  resolved,  is  therefore  independent  of  the  others,  and 
its  action  may  be  determined  upon  the  principles  laid  down  in 
§  46. 

The  general  equation  of  variable  motion  (56)  is  therefore  ap- 
plicable to  this  case  ;  and  calling  the  three  rectangular  forces  X, 
Y  and  Z,  and  the  spaces  in  these  directions,  #,  y  and  z,  it  will  be- 
come 


These  equations  contain  all  that  it  is  necessary  to  know  in  respect 
to  the  motion.     For  1st,  by  an  integration  we  obtain  the  values 
of  the  three  velocities  in  these  directions  ;  these  from  (55)  are 
dx        dy        dz.  ,     . 

di'     dt1     dt'  (63) 

from  the  composition  of  which,  the  velocity  of  the  moving  body  is 
determined; 

2.  Another  integration  gives  us  the  co-ordinates  in  terms  of 
the  time  : 

3.  By  eliminating  £,  we  obtain  the  equations  of  the  curve  that 
the  body  describes. 

In  the  same  case,  of  the  resolution  of  all  the  forces  into  three 
rectangular  forces,  we  shall  have 

(64) 


Book  II.}  OF  MOTION.  51 

for,  from  the  above  equations  we  .obtain,  by  multiplying  them 
respectively  by  dx,  dy,  dz,  and  adding  them  together, 

dxtfx+dytfy+dztfz^ 


but  as  the  axes  are  rectangular, 

whence  we  obtain 

dxd3x  +  dyd2y  +  dzd2  z  =  dsd?s  ; 
therefore 

dsd2s     ds         ds 
z±=~-=       d.       -~ 


If  the  forces  act  in  one  plane,  we  have  need  only  of  the  two 
equations, 


If  but  two  forces,  acting  in  the  same  plane,  are  concerned,  it 
may  be  simpler  to  employ  them,  without  resolving  them  each  into 
three  parallel  to  three  rectangular  axes  ;  the  same  principles  ap- 
ply to  them  as  to  three  rectangular  forces,  and  the  equation  (56) 
becomes,  calling  the  forces  F  and  F', 


When  no  more  than  two  forces  act,  as  the  successive  diagonals 
are  all  in  one  plane,  the  curve  is  a  plane  curve, 


58  OF  MOTION.  [Eouh  It 

CHAPTER  IV. 
OF  PARABOLIC  MOTION. 

53.  If  a  point  be  acted  upon  by  two  forces  not  in  the  same  straight 
line,  by  virtue  of  one  of  which,  it  would  describe  a  straight  line 
with  uniform  velocity,  and  of  the  other,  a  straight  line  with 
uniformly  accelerated  velocity;  and  if  the  second  force  act  paral- 
lel to  itself,  during  the  continuance  of  the  motion;  the  point  will 
describe  a  parabola  under  their  joint  action.  The  diameters  of 
this  parabola  are  parallel  to  the  directions  of  the  accelerating 
force,  and  its  parameter  is  four  times  the  space  due  to  the  velo- 
city communicated  by  the  first  of  the  forces. 

Let  F'  be  the  projectile  force  by  whose  action  the  body  would 
go  on  with  uniform  rectilineal  velocity,  and  F  the  accelerating 
force  which  begins  as  soon  as  the  force  F'  ceases  to  act,  then 
from  equation  (66) 


As  F  represents  the  accelerating  force,  call 

V=g; 

then,  F'  representing  the  force  that  produces  the  constant  velocity, 

F'=0; 
hence 

df  df 

piMtfi  °=*-JT 

Integrating 


A  and  B  representing  the  initial  velocities  in  the  two  directions  ; 
but  as  the  accelerating  force  begins  to  act  at  the  instant  whence 
the  motion  is  to  be  computed, 

A=0. 

And  we  may  measure  B  in  terms  of  g-,  and  the  space  through 
which  the  body  should  pass  under  its  action,  in  order  to  acquire 
the  initial  velocity  ;  by  (61) 


substituting  these  values,  we  have 
df          df 


Book  II] 

Integrating  anew 


OP    MOTION. 


53 


In  which  expressions  the  constant  quantities  are  omitted,  for  <=0, 
/=0,  f=Q,  at  the  same  time ; 
eliminating  t 

f'*=4sf  (67) 

which  is  the  equation  of  a  parabola,  whose  ordinate  and  abscissa 
are/'  and  /,  and  whose  parameter  is  4s.  /  being  the  abscissa, 
shows  that  the  diameters  of  the  parabola,  and  of  course  its  axis, 
are  parallel  to  the  direction  of  the  accelerating  force. 

To  investigate  the  path  by  means  of  two  rectangular  co-ordi- 
nates, X  and  Y,  the  directions  of  X  being  parallel  to  the  direction 
of  the  accelerating  force. 

In  the  figure  beneath,  let  AP  represent  the  action  of  the  accele- 


rating force,  AT  the  direction  in  which  the  projectile  force  would 
cause  the  body  to  move  with  uniform  velocity. 

Draw  AQ  to  represent  the  direction  of  the  axis  X  ;  TR  parallel 
to  AP ;  and  PR  parallel  to  AQ  ;  join  PM,  then 
AQ=:r,  QM=7/,  MR=i/-r-/; 
call  the  L  TAQ=  L  MPR  ,  t, 
then  !/+/—/'  sin.  t,  x=f  cos.  i ; 

whence 

x  sin.  t 


but  from  (67) 
therefore 


<•=£.    - 


45      4*  cos. 


tan.  i — — 


4a  cos.  s  i  ' 


(68) 


1t  OF    MOTIMW.  [Bouh  II 

CHAPTER  IV. 
Or  PARABOLIC  Morion. 

53.  If  •  point  bo  acted  upon  by  two  forces  not  in  tbe  same  straight 
line,  by  virtue  of  one  of  which,  it  would  describe  a  straight  line 
with  uniform  velocity,  and  of  the  other,  a  straight  line  with 
uniformly  accelerated  velocity;  and  if  the  second  force  act  paral- 
lel to  iUelf,  during  the  continuance  of  the  motion;  MM  point  will 
describe  a  parabola  under  i  !  ><i  i  joint  action.  The  diameters  of 
this  parabola  are  parallel  to  the  directions  of  the  accelerating 
force,  and  its  parameter  is  four  times  the  space  due  to  the  vH., 
city  communicated  hy  the  first  of  the  forces. 

Let  F'  be  tbo  projectile  force  by  whose  action  the  body  w.ul«l 

go  on  with  UIHI'MIHI  roctilinftal  velocity,  and  F  the  acccl<  i  .MI.,^ 

-•  In.  l>  U'giriM  us  noon  an  the  force  F'  ceases  to  act,  then 

from  .,|,i.  ,(,.,,,  'Mi; 

<ir   '•'*    -. 

As  F  represent*  th«  aceoleroting  force,  call 

V     K; 
ii>  •  n,  F'  representing  the  force  that  produces  the  constant  velocity, 

I  0; 


Integra  ting 


A  and  B  representing  the  initial  velocities  in  il.«  iu,,  din  -.  -HMHI  , 
but  as  the  accelerating  force  begins  to  act  at  the  mitnm  uh.  n«, 
the  motion  is  to  be  computed, 

A«0. 
A  n<  I  we  may  measure  B  in  terms  of  #,  and  the  space  through 

\vlnrl,  )!,.•  |.,..lv  slioul.l  |,;,SM  ,„..!.,    ,|  ,    ,,,-1  ......   u.  oidrr  to  ar«|uiru 

the  initial  velocity  t  by  (61) 


substituting  these  flfaet,  we  have 

•If         df 

.1,    **<   ,// 


JBtwk  11] 


OF 


53 


111  \\hirh  expressions  (In-  rmislnill  (]iiimlilii>s  lire  oiniltfd,  for  f      0, 

/•"O,  /=0,  at  the  iam<<  limr ; 

climmiilin;',  / 

/"=4</  (67) 

Wllicll  is  llir  ••.|!i.i!:..n   . -I    ;i   pimiholll,     \\hoc  nidin.ili-   .Mill  llllHCINHn 

arc/'  aiul ./,  mid  \vli<..^«-  |»;II,HIU  id  is  is.     /  liriu^r  iho  iibrtciaiaf 

N|,(,\\;I  thill   Ihr  ili.'inirlrr.s  ol'lh,-  |i;ir:il>iiln,    ;m.l  o|  roin.si-  ll  -,    u\i    . 
n n-  [>!irjlll<ll  lo  lh<'  iliHM-lion  ofllic  nrrcli-cililiH  loH'f. 

To  in\«- •  li".ilr  III.-  |.:ilh  hv  iiirinr;  ul'luu  n-clni|i>-iiliir  ro-ordi- 
lialrii,  \  ..11,1  \  ,  Ihr  (liiiTlmiis  ni'  \  linii>>  |>lllllll<-l  In  llir  dilTrlloil 
of  !hn  iicci-lrinliut'  I'OK  ••. 

In  tlio  liguro  bcnuuth,  lot  AP  roprenont  the  action  of  thes  accele- 


rating force,  AT  the  direction  in  which  the  projectile  force  would 
cause  the  body  to  move  with  uniform  velocity. 

Dl.'.u     \<J  lo  irpiv.srill   ihrdiMTlion  of  llir  ;IM      \    ,     I   K  ,,.,,.  ,11.  i 
1<>  AC;   mid  Pit  pimdlnl  (<>    \o  ,    |<>IM  I'M,  thru 

AQ=j?,  QM=y,  MR=t/4-/; 
call  the  /lTAQ=^MPRftf 
then  !/+/=/'  "in.  i,  »=/'  coi.  i  \ 

\\h«-nco 

x  «in.  i 


I.NI  IVi.m  (Ii7) 
(hnrloro 


/-£- 


4f  ooi.  »i  ' 

V"; 


54  op  MOTION.  [Book  II. 

This  last  equation  may  be  obtained  directly,  from  the  resolu- 
tion of  the  two  forces  into  two  rectangular  forces,  one  of  which 
is  parallel  to  the  direction  of  the  accelerating  force. 

Call  the  measure  of  the  accelerating  force  g  ;  as  it  acts  in  a 
direction  opposite  to  the  co-ordinate  Y,  it  will  be  negative,  or  —  g. 

The  alteration  in  the  direction  of  X,  being  wholly  due  to  the 
force  that  produces  a  constant  velocity,  is  =0. 

The  two  equations  (62)  become 

d2x_         d2y_ 
dT2"0'      ~dT~S' 
Integrating, 

y=—  ^-+Cf+c',  x=bt+b'.  (69) 

Taking  the  origin  of  the  co-ordinates  at  the  point  A,  when  /=0, 
*=0,  i/—  0,  therefore  6'=0,  c'=0. 

Let  v  be  the  velocity  due  to  the  projectile  force,  or  the  initial 
velocity,  call  the  angle  TAQ,  as  before,  i.  The  components  of 
the  initial  velocity  are  v  cos.  t,  acting  in  the  direction  of  X  ;  and 
v  sin.  f,  acting  in  the  direction  ofy  ;  and  from  (63) 

dx  dy 

—  =v  Cos.  t,    ji=v  sin.  »  ; 

and  as  the  constant  quantities,  6  and  c,  represent  these  velocities, 

b=v  cos.  «',     c—v  cos.  »; 
therefore  by  (69) 


gt2 
y=  —  -g  —  \-vt  sin.  t,    x=vt  cos.  i  ; 


eliminating  /, 


u=x  tan.  t  —  r~r  -  r~-  ? 
2v2  cos.2  t  ' 

but  as  v—\/2gs  (61),  t?2=2g-5,  by  the  substitution  of  which  we 
obtain, 


A  comparison  of  this  with  (67),  by  means  of  the  preceding  in- 
vestigation, shows  that  this  is  the  equation  of  a  parabola,  whose 
axis  is  parallel  to  the  direction  of  the  accelerating  force. 

The  maximum  value  of  g  is  VS,  the  length  of  the  axis  of  the 
parabola.  In  this  case  cfa/=0,  and  the  differential  of  (68)  is 

xdx 


tan.  t.  cos.  t==sin.  t, 

x=2a  sin.  t  cos.  «  ; 


Book  II.  ]  OP  MOTION.  55 

and  as,  2  sin.  i  cos.  «=sin.  2  t, 

x=s  sin.  2t;  (70) 

substituting  this  value  of  x  in  (68),  we  obtain 

y=SSm.*i.  (71) 

The  value  of  x  (70)  is  equal  to  AS  ;  this  is  half  of  AB,  which 
we  shall  call  A ;  therefore 

A=2s  sin.  2«.  (72) 


56  or  MOTION.  [Book  11. 


CHAPTER  V. 

OF    THE    MOTION    OF    POINTS    COMPELLED    TO    MOVE    UPON    SURFACES, 
UNDER    THE    ACTION    OF    ACCELERATING    FORCES. 

54.  When  a  point  rests  upon  a  given  surface,  this  surface  may 
be  considered  as  exerting  a  force  to  resist  the  pressure,  which 
force  acts  in  the  direction  of  a  normal  to  the  surface.     In  the  case 
of  equilibrium,  this  is  equal  to  the  resultant  of  all  the  other  forces, 
§  19.     But  in  the  case  of  motion,  the  action  of  the  surface  must 
be  found,  by  decomposing  the  resultant  of  all  the  forces  that  act 
into  two  components,  one  of  which  is  parallel,  the  other  a  normal 
to  the  surface.     The  last  of  these  will  represent  the  intensity  with 
which  the  point  is  pressed  against  the  surface,  by  the  forces,  and 
will  be  in  equilibrio  with  the  resistance  of  the  surface  ;  the  for- 
mer therefore,  will  alone  remain  to  cause  the  motion  of  the  body. 

55.  The  simplest  case  that  can  occur  is  where  the  surface  is 
plane,  and  is  inclined  at  a  given  angle  to  the  direction  of  an  ac- 
celerating force,  which  is  always  parallel  to  itself.  In  this  case,  the 
point  that  is  impelled  by  the  force  will  describe  a  straight  line, 
which  is  the  common  intersection  of  the  surface  and  the  plane  in 
which  the  accelerating  force  acts.     So  also  if  a  similar  force  act 
upon  a  point  placed  on  a  curve  surface,  the  point  will  describe  a 
plane  curve,  which  is  the  common  intersection  of  the  surface  and 
the  plane  in  which  the  force  acts.     In  these  instances  the  surfaces 
may  be  considered  as  straight  lines  or  plane  curves.  So  also  when 
the  force  is  constantly  directed  to  a  point,  and  its  directions  all 
lie  in  one  plane,  the  curve  described  is  a  plane  curve.     In  these 
several  instances,  the  circumstances  of  the  motion  may  be  inves- 
tigated in  a  more  elementary  manner  than  can  be  done  by  pur- 
suing the  general  method  of  curvilinear  motion,  and  referring  the 
forces  to  their  rectangular  co-ordinates.     When  the  point  moves 
on  a  plane  surface,  we  shall  continue  to  use  the  name  of  the  sur- 
face ;  but  in  all  other  instances,  we  shall  merely  name  the  curve 
that  is  described.     In  our  subsequent  applications  of  the  theory, 
the  modes,  in  which  the  path  of  a  body  may  be  made  to  coincide 
with  the  several  curves,  will  be  pointed  out. 

56.  When  a  point  moves  upon  a  plane  surface,  under  the  action 
of  an  accelerating  force  whose  intensity  is  constant,  and  whose 
direction  is  always  parallel  to  itself,  that  part  of  the  accelerating 
force  which  remains  to  cause  the  motion  of  the  point,  is  also  a 
constant  accelerating  force,  and  is  to  the  whole  accelerating  force 


Book  II :\ 


OF    MOTION- 


in  the  ratio  of  the  cosine  of  the  inclination  of  the  surface  to  the 
direction  in  which  the  point,  if  free,  would  move  under  the  action 
of  that  force. 

Let  A  be  the  point,  moving  upon  a  plane  surface,  CD,  under 
the  action  of  a  constant  force  g-,  whose  direction  makes  with  that 
of  the  surface  the  angle  i'.  Draw  BC  perpendicular  to  BD,  which 
is  parallel  to  the  directions  of  the  accelerating  force,  the  length 
of  the  plane  being  CD,  we  shall  call  BD  its  height. 


Decompose  the  force  g  into  two,  one  of  which,  p,  is  a  normal 
to  the  surface,  the  other,/ is  parallel  to  it ;  their  values  will  be  (14) 

p=g.  sin.  t',   /—  g  cos.  i'. 

The  action  of  the  former  is  destroyed  by  the  action  of  the  sur- 
face, the  latter  remains  to  move  the  point  along  it. 

The  motion  of  the  point  along  the  surface  will  be  uniformly  ac- 
celerated, for /bears  a  constant  relation  to  g-,  which  is  a  constant 
accelerating  force. 

All  the  formulae  in  §  49,  that  have  reference  to  uniformly  ac- 
celerated motions,  are  applicable  to  this  case,  by  substituting/, 
or  its  value  g  cos.  t ,  for  g ;  hence 

v'=gt  cos.  e", 

i) 
8'=7r- 


~2g  cos.  t' 
gt 2  cos.  i' 


t'— 


(73) 


g  cos.  * 

Comparing  these  with  the  quantities  of  the  same  kind  in  free  mo- 
tion, under  the  action  of  the  same  accelerating  force,  we  have  for 
the  ratio  of  the  velocities  acquired  in  equal  times, 


— =cos. 
v 


(74) 


53  or  MOTION.  [Book  //. 

for  the  ratio  of  the  spaces  described  in  equal  times, 

—  =cos.  t';  (75) 

s 

for  the  ratio  of  the  spaces  described  in  attaining  equal  velocities, 
-r=cos.  i"  ;  (76) 


for  the  ratio  of  the  times  in  which  equal  velocities  are  attained, 
y=cos.  t;  (77) 

the  velocity,  acquired  in  passing  freely  through  the  height  of  the 
plane,  is  (61) 


in  moving  along  the  plane  surface, 

t/=V(2gcos.  i'-^s77') 

these  velocities  are  therefore  equal,  or 

v=v'.  (78) 

In  plane  surfaces  of  unequal  inclinations,  but  equal  heights, 
the  velocities  attained  are  therefore  equal. 

If  the  surface  coincide  in  direction  with  the  chord  of  a  circle, 
that  terminates  at  either  end  of  the  diameter  that  corresponds 
with  the  direction  of  the  force,  the  time  of  describing  the  surface, 
and  passing  freely  through  the  diameter,  will  be  equal. 

Call  the  diameter  c, 

2<z 

.      '=^1P 

the  length  of  the  chord  is  a  cos.  t  ,  and  the  time  of  describing  it, 
2a  cos.  t'          2a 

'-'T^r^T-  (79) 

And  the  same  will  be  true  of  all  chords,  terminating  at  either  ex- 
tremity of  the  diameter  which  is  parallel  to  the  accelerating  force. 
The  time  of  describing  planes  of  equal  inclination,  is  as  the 
square  roots  of  their  lengths.  Call  the  lengths  I  and  I'  ;  the  times 
/'  and  t", 

21  21' 

t'=V  --  •  ,    t"=V  --  ., 

g  COS.  I  g  COS.  I 

21  21' 

"  tf   '    4ff       •       /  / 

".  r  .  i    '  •  v  ~i  '  v  •'  » 

g-cos.1          gees,  t 

/'  :  t"  :  ;  Jl  :  VI',  (80) 


Book  //.] 


OF    MOTION. 


59 


57.  If  a  point  move  from  rest,  upon  a  system  of  plane  surfaces,, 
under  the  action  of  a  constant  accelerating  force,  it  will  acquire 
the  same  velocity  as  it  would  have  acquired,  in  moving  under  the 
action  of  the  same  force,  upon  a  single  plane  surface  of  a  height 
equal  to  that  of  the  system  ;  or  to  that  it  would  have  acquired,  in 
moving  freely  through  the  height  of  the  system. 

Suppose  firs  t,  that  there  are  two  planes,  AB,  BC  ;  let  AE  be 

D  S 


a  line  perpendicular  to  the  direction  of 'the  accelerating  force ; 
produce  the  plane  BC,  until  it  intersect  AE  in  D.  A  point  moving 
from  a  state  of  rest  under  the  action  of  the  force,  along  AB,  will 
acquire  the  same  velocity  as  in  moving  freely  through  DF,  or  in 
moving  along  a  surface  in  the  direction  BD  ;  it  will  therefore  go 
on  in  BC  as  if  it  had  come  from  D,  and  will  reach  C  with  the  ve- 
locity it  would  have  acquired  in  describing  the  plane  DC,  or  in 
passing  freely  through  DG. 

Had  there  been  three  planes  in  the  system,  the  same  mode  of 
reasoning  would  have  led  to  the  same  result,  and  so  for  any  num- 
ber, or  for  a  curved  surface. 

58.  In  this  proposition,  it  is  obvious  that  any  resistance  that 
might  take  place  at  the  angles  of  the  planes,  is  left  out  of  account. 
If  the  planes  in  the  system  become   infinitely  small,  the  surface 
becomes  curved,  and  this  proposition,  therefore,  holds  true  in  res- 
pect to  the  arc  of  a  curve,  in  which  the  acquired  velocity  is  the 
same  as  that  acquired  in  moving  freely  through  its  height. 

59.  A  point  moving  in  a  cycloid,  under  the  action  of  a  con- 
stant force  describes  every  different  arc  that  is  terminated  at  the 
extremity  of  the  diameter  of  the  generating  circle,  in  equal  times  ; 
and  the  time  of  describing  each,   is  to  the  time  of  free  motion, 
through  the  diameter  of  the  generating  circle,  as  half  the  circum- 
ference of  a  circle  is  to  its  diameter. 


60 


OF    MOTION. 


[Book  II. 


If  the  constant  force  g-,  that  acts  at  the  point  C  of  the  curve, 
be  resolved  into  two  components,  one  of  which  is  parallel,  the 

JL _*' 


other  perpendicular  to  the  tangent  at  that  place  ;  as  the  tangent 
will  coincide  with  the  element  of  the  curve  ds,  that  part  which  re- 
mains to  cause  the  motion  in  the  curve,  will  be  to  the  whole  accele- 
rating force,  as  CB  to  DB.  But  of  these,  DB  is  a  constant  quan- 
tity, and  BC  is  half  of  the  arc  CE.  Hence  the  force  that  acts 
at  any  part  of  the  curve,  is  proportioned  to  its  distance  from  the 
lowest  point.  Now  if  two  different  arcs  of  the  curve,  both  ter- 
minating at  the  point  E,  be  supposed  to  be  divided  into  an  equal 
number  of  elements,  in  each  of  which  the  velocity  is  constant  ; 
the  forces  will  be  proportioned  to  these  elements,  and  they  will 
in  consequence  be  described  in  each  case  in  equal  times  ;  and  as 
the  number  of  elements  are  equal,  the  sums  of  these  times  will 
be  equal  also,  and  both  arcs  will  be  described  in  equal  times. 
The  whole  semicycloid,  FE,  will  therefore  be  described  in  the 
same  time  with  any  of  its  smallest  arcs. 

The  time  of  describing  the  semicycloid  is  easily  found  from 
the  general  expression  of  variable  motion,  (53), 

ds 


hence 


ds 


(81) 

but  for  the  semicycloid,  the  diameter  of  whose  generating  circle 
is  a, 


ive  therefore  obtain  for  the  integral  of  this  expression  (81) 

t=l<S~g  (82) 

*  being  the  ratio  of  the  circumference  of  a  circle  to  its  diameter. 

The  time  of  the  fall  through  the  radius  of  the  generating  circle  is 

a  tjf 

V~ ;  to  which  the  last  value  of  t  has  the  ratio  of  ~  :  1. 

o  ^ 


J&00k  //]  OF    MOTIOlf.  61 

60.  If  the  body  describe  a  circular  arc  of  very  small  size,  the 
time  of  describing  it  is  the  same  as  that  in  which  an  arc  of  a  cy- 
cloid is  described,  if  both  terminate  at  the  point  where  the  direc- 
tion of  the  force  is  perpendicular  to  the  curve  ;  but  if  the  arcs  have 
any  amplitude,  the  time  of  describing  the  circular  arc  will  be 
greater  than  that  of  describing  the  arc  of  a  cycloid,  by  the  quan- 
tity —  ;  in  which  h  is  the  versed  sine  of  the  arc  described,  and 
a  the  radius  of  the  circle. 

Suppose  the  body  began  to  move  from  P,  under  the  action  of 
an  accelerating  force,  acting  parallel  to  the  radius  C  A,  and  to  have 
reached  the  point  M. 


Let  AR=#  ;  RM=t/  ;  AM=s  ;  the  radius  CA—  a,  and  AS=/s  ; 
we  have  for  the  point  M,  from  the  nature  of  the  circle, 

adx 


and  the  velocity  at  M  being  due  to  the  height  SR, 


g  being  the  measure  of  the  constant  accelerating  force  ;  hence 
ds        a  dx 


'  V(h—x).(2ax—  x2)  ' 
which  can  only  be  integrated  by  means  of  a  series.  For  this  pur- 
pose the  equation  is  resolved  into  the  following  form,* 


dt=: 


2a 


2V g'  V(hx-x2) 
which  developed,  and  integrated  from  x=h  to  #=0,  gives, 


See  Venturoli,  Porsson,  and  Laplaae. 


OP   MOTION.  [Book  II. 

z    h         1.32     h? 

+&c--  <83> 


In  small  arcs'  the  first  term  of  the  series  is  alone  necessary,  and 
the  expression  becomes 

'=5^  x  0+15)5 

in  very  small  arcs  the  series  vanishes  altogether,  and 

<=|V|.  (84  a) 

61.  When  the  accelerating  force,  instead  of  acting  parallel  to 
itself,  is  directed  to  a  fixed  point,  it  is  called  a  central  force. 

It  will  be  obvious  from  what  has  already  been  said,  thbt  a 
point  acted  upon  by  a  projectile  force,  and  afterwards  drawn  to- 
wards a  centre  by  an  accelerating  force,  must  describe  a  curve. 

The  curve  thus  described  is  called  a  Trajectory,  or  Orbit. 

If  the  point  after  moving  completely  around  the  centre  of  force, 
again  describe  the  same  path,  the  orbit  is  said  to  be  re-entering. 

The  time  in  which  a  re-entering  orbit  is  described,  counting 
from  the  instant  in  which  the  moving  point  sets  out  from  a  given 
position  in  the  curve,  until  it  return  to  it  again,  is  called  the 
Periodic  Time. 

A  line  drawn  from  any  point  in  the  orbit  is  called  a  Radius 
Vector. 

62.  The  simplest  case  of  central  force  is,  where  a  body,  con- 
nected to  a  fixed  point  by  an  inflexible  straight  line,  is   impelled 
by  a  projectile  force  at  right  angles  to  that  line.     The  latter  force 
would  have  impressed  upon  the  body,  a  motion  with  an  uniform 
velocity.     The  body  will  then,  in  consequence  of  its  connexion 
with  the  fixed  point,  describe  a  circle,  of  which  that  point  is  the 
centre.  If  the  connexion  were  to  cease  at  any  point  in  the  curve, 
the  deflecting  force  would  cease  to  act,  and  the  body  would  go  in 
a  straight  line,  whose  direction  would  be  a  tangent  to  the  curve. 
The  force  acting  at  any  point  in  the  curve,  must  therefore  be  decom- 
posed into  two,  one  of  which  is  in  the  direction  of  the  curve,  the 
other  in  that  of  its  radius.     The  last  of  these  is  called  the  centri- 
fugal force,  and  is  equal  and  directly  opposite  to  the  force  that 
draws  the  body  towards  the  centre,  and  which,  for  distinction 
sake,  is  called  the  centripetal  force. 

In  elementary  treatises,  the  consideration  of  the  action  of  cen- 
tral forces,  is  usually  limited  to  the  case  of  motions  in  circular 
orbits,  described  under  circumstances  analogous  to  those  we  have 
recited.  If  instead  of  an  inextensible  straight  line,  an  attractive 
force  equal  and  directly  opposed  to  the  centrifugal  force  that  has 


Book  If.]  OP  MOTION.  63 

just  been  defined,  be  substituted,  the  circumstances  will  be  the 
same  as  if  the  connexion  were  made  by  theinextensible  line,  and 
the  body  will  in  like  manner  describe  a  circle.  The  relations 
among  the  forces,  velocities,  times,  and  spaces,  that  are  found  in 
respect  to  circles,  may  be  applied  to  the  case  of  other  curves;  for 
a  motion  in  a  curve  may,  for  a  short  space  of  time,  be  considered 
as  corresponding  with  that  in  a  circle,  whose  radius  is  the  radius 
of  curvature  of  that  part  of  the  curve. 

63.  In  describing  a  circular  orbit,  under  the  action  of  two 
forces,  one  of  which  is  projectile,  the  other  constant,  and  directed 
to  the  centre  of  the  circle,  a  point  will  move  with  uniform  ve- 
locity. 

Let  the  point  be  situated  in  the  curve  at  A,  and  suppose  a  pro- 


G 


jectile  force  to  act,  which  in  an  infinitely  small  time,  would  carry 
it  forward  in  the  direction  AB,  to  the  point  B  ;  the  central  force 
would  in  the  same  time  cause  a  deflection  to  D.  If  the  central 
force  were  not  to  cease  to  act,  the  point  would  go  on  in  the  direc- 
tion AD  produced,  and  describe  in  an  equal  portion  of  time  a 
space  DE  equal  to  AD  ;  but  the  central  force  continuing  to  act, 
brings  it,  at  the  end  of  an  equal  interval,  to  the  point  F  ;  and  in 
this  way  it  would  be  found  at  the  end  of  the  third  interval  at  H. 
Now  it  is  obvious  that  the-  several  triangular  spaces  CAB,  CAD, 
CDE,  CDF,  CEG,  CFH,  are  all  equal,  and  hence  the  radius 
vector  describes  equal  areas  in  equal  times ;  these  areas  are  how- 
ever in  fact,  sectors  of  the  curve  ;  and  in  a  circle,  whose  radii  are 
all  equal,  if  the  centre  be  also  the  centre  of  force,  the  arcs  which 
are  described  by  the  moving  point,  are  also  equal  in  equal  times  ; 
in  uneqiial  times,  proportioned  to  the  times  ;  and  the  circle  is  de- 
scribed with  uniform  velocity. 

64.  In  circular  orbits,  the  force  is  represented  by  the  square 
of  the  velocity  divided  by  the  radius. 

The  projectile  velocity  is  such  as  would  have  been  acquired 
by  falling  freely,  under  the  action  of  the  central  force,  through 
half  the  radius  of  the  circle. 


64 


OP    MOTION. 


[Book  II. 


If  points  revolve  in  different  circles,  the  central  forces  are  di- 
rectly as  the  radii  of  the  orbits,  and  inversely,  as  the  squares  of 
the  periodic  times ;  if  the  times  be  equal,  they  are  therefore  as 
the  radii ;  if  the  radii  be  equal,  they  are  inversely  as  the  squares 
of  the  periodic  times. 

Let  a  point  describe  a  very  small  arc,  AB,  in  the  element  of 
C  the  time  dt.  During  this  small  period,  the  versed 
sine,  AD,  will  be  the  space  described  under 
the  action  of  the  central  force ;  this  force  may 
therefore  be  measured  by  the  velocity  that  would 
have  been  acquired  in  moving  through  AD  with 
accelerated  motion;  the  formula  (61)  gives 

2s  2s 

t?=-^-,  whence  f—~p* 

In  this  expression,  2*  is  twice  the  versed  sine  of 
AB  ;  the  very  small  arc,  AB,  may  be  consider- 
ed as  coinciding  with  its  chord,  and  the  chord 

is  a  mean  proportional  between  the  radius  and  the  versed  sine  ; 

call  the  arc  AB,  o,  and  we  have 

2  a3      a2 
2s=- 


2r 


and 


now  as  the  velocity  in  the  circle  is  uniform, 


whence 


»=-j  and  v*=(-.)   ; 


(85a) 


/=v  .  (85) 

To  compare  the  intensity  of  this  force  with  that  of  the  central 
force  g,  supposed  to  be  constant ;  from  (61) 

v2=gs, 
whence 


(86) 


and  when /is  equal  to  g, 


the  space  then  to  which  the  velocity  is  due,  is  equal  to  half  the 
radius. 


Book  II.]  or  MOTION.  65 

Let  T  be  the  periodic  time,  the  circumference  of  the  circle  is 
«r,  and  (85a) 

•  ==~T~ '  (87) 

substituting  this  value  of  v  in  that  of/,  we  have 

(88) 

•whence  the  forces  are  directly  as  the  radii  of  the  orbits,  and  in- 
versely, as  the  squares  of  the  periodic  times. 

65.  If  the  central  force  be  such  as  varies  in  the  inverse  ratio  of 
the  squares  of  the  distances,  the  squares  of  the  periodic  times  are 
proportioned  to  the  cubes  of  the  distances. 

If  two  forces,  F  and  /,  act  in  different  circles,  with  intensi- 
ties inversely  proportioned  to  the  squares  of  the  distances,  by  (85) 
V2  v2 

F==RT'    /=T'  and 

F:/:  :  V2  :  v2  ; 
but  by  hypothesis, 

F:/::^:^; 

therefore 

V2     u3 

.    .    .  «2  .    R2  . 

R  '    r    ' 
whence 

V2:  v2:  :r3:R3; 
and  as  the  times  are  inversely  as  the  velocities, 

T2  :  i3  :  :  R3  :  r3,  (89) 

or  the  squares  of  the  periodic  times,  are  as  the  cubes  of  the  radii. 

66.  Having  thus  investigated  the  circumstances  of  motion,  in 
those  lines  and  curves  that  will  be  of  most  frequent  application 
in  practice,  by  more  elementary  processes  ;  we  shall  next  exhi- 
bit the  general  principles  that  are  applicable  to  any  case  what- 
soever, by  means  of  the  resolution  of  all  the  forces  into  three, 
parallel  to  three  rectangular  co-ordinates. 

Call  the  resistance  of  the  surface  N;  the  angles  that  the  rectan- 
gular axes  make  with  the  normal  to  the  surface  a,  6,  c  ;  the  com- 
ponents of  the  resistance  will  be  (18) 

N  cos.  a,     N  cos.  6,     N  cos.  c ;  (90) 

the  values  of  the  three  rectangular  components  of  the  active 
forces  are  by  (62) 

v_^£.'Y_^     7-^ 
*-~dP  '    *~  dP  '   *~  df" 


65  or  MOTION.  [Book  II. 

to  each  of  which  must  be  added  the  component  of  N  that  acts  in 
its  direction ;  whence  we  have,  for  the  equations  of  motion  upon 
a  curve, 


X-f-N  cos.  a— ~~£p  ; 


Y+N 


cos.    = 


Z-f  N  cos.  c—- 


(91) 


the  values  of  the  velocities  from  (63),  are 

dx         dy         dz 

dt'       dt'       df> 

these  may  be  reduced  to  the  direction  of  the  normal,  by  multi- 
plying each  by  the  cosine  of  the  angle  it  makes  with  that  line  ; 
and  as  the  direction  of  the  motion  is  at  right  angles  with  the  nor- 
mal, the  sum  of  these  three  components  must  be  0,  or 

—  cos.  «+ 37  cos.  6-f—  cos.  c=0.  (92) 

dt  dt  dt 

We  may  eliminate  the  unknown  quantities  in  the  three  equa- 
tions, (91)  by  adding  them  together,  after  multiplying  the  first  by 
dx,  the  second  by  dy,  and  the  third  by  dz  ;  the  sum  of  this  addi- 
tion becomes,  if  we  take  into  account  the  equation  (92), 

"^y^^MH-TdH-Zfe  (93) 

If  the  quantity  Xcta+Xcfy-t-Xcte,  is  the  exact  differential  of  the 
three  variable  quantities,  x,  y,  z,  we  obtain  by  integrating 

*tbJ+*?=1/(,,,^  +  C,  (95) 

dx     dii     dz 
now  as  —     ~f    — ,  are  the  components  of  the  velocity  in  the 

direction  of  the  three  axes  ;  the  first  member  of  this  expression 
represents  the  square  of  the  velocity,  and 

v*=2f(x  ,  y  ,  z}.  (96) 

It  may  be  remarked  that  there  are  several  cases  in  practice, 
where  the  expression  Xdx-\-Ydy -\-Zdz,  is  not  an  exacf  differential 
of  three  variable  quantities ;  such  for  instance  is  the  case,  where 
a  body  is  acted  upon  by  a  fluid  resistance,  or  by  friction. 

Where  the  three  rectangular  forces,  X,  Y,  and  Z,  are  each 
equal  to  0, 

/>,?/,  z)  =  0, 
and 

t>3=C, 
or  the  square  of  the  velocity,  and  consequently  the  velocity  itself, 


Baok  II.]  oy  MOTION.  ($ 

is  constant.  Thus,  then,  when  the  accelerating  force  ceases  to 
act,  the  body  will  continue  to  move  with  uniform  velocity. 

.If  the  forces,  X,  Y,  and  Z,  retain  a  determinate  magnitude,  the 
velocity  is  no  longer  constant,  but  it  is  independent  of  the  curve 
the  body  is  compelled  to  describe ;  for  if  we  call  A  the  velocity 
that  corresponds  to  another  point  of  the  curve,  whose  co-ordinates 
are,  a,  6,  c;  we  can  infer  the  same  in  respect  to  this,  as  to  the  ori- 
ginal case,  and 

A2=C+2/(a  ,  6  ,  c), 
which  substracted  from  the  former  equation,  gives 

v*—A2=2f(x  ,y,z,  )— 2/(a  ,  6  ,  c)  ;  (97) 

it  is  obvious  therefore,  that  the  change  in  the  squares  of  the  velo- 
cities, has  no  relation  to  the  form  of  the  curve  described  between 
the  two  points.  So  also,  when  several  bodies  set  off  from  a  given 
point,  under  the  action  of  the  same  accelerating  force,  they  will  all, 
on  reaching  another  point,  have  the  same  velocities,  however  vari- 
ous may  be  the  lines  described  in  the  mean  time,  and  different  the 
times  of  describing  them.  This  is  an  obvious  generalization  of 
what  was  found,  in  §  58,  to  happen,  when  the  motion  took  place  in 
plane  curves,  under  the  action  of  a  constant  force. 

The  further  pursuit  of  this  subject,  by  higher  methods  of  analy- 
sis, leads  to  a  remarkable  law  which  is  called  The  Principle  of 
the  Least  Action.  It  may  be  thus  expressed : 

When  a  body  is  acted  upon  by  forces  whose  relation  is  expressed 
by  the  formula,  (95)  and  the  curve  is  not  determined,  it  will  choose 
for  the  direction  of  its  motion  the  shortest  line  that  can  be  drawn 
on  the  surface,  or  that  in  which  the  integral,  fvds,  is  a  minimum. 

When  the  accelerating  force  ceases  to  act,  the  velocity  is,  as 
we  have  seen,  constant,  and 

fvds=vs, 

and  as  s  is  the  shortest  space,  the  body  will  pass  from  one  point 
to  another,  in  the  shortest  possible  time. 

For  the  further  illustration  of  this  subject,  we  refer  to  Bowditch's 
translation  of  Laplace. 

67.  When  a  point  is  acted  upon  by  but  one  accelerating  force, 
constantly  directed  towards  a  fixed  point,  the  path  will  be  a  plane 
curve;  and  the  areas,  described  around  this  point  by  the  radius 
vector,  are  proportioned  to  the  times  employed  in  describing 
them. 

By  (62), 

^_Y     <^_  #Z 

W  '    df~        '    ~d?~* 

these  are  readily  transformed  into  the  following 

— 7/X)  .  dt* 
—xZ}  .  d? 
—zY  .  df. 


68  OP    MOTION.  [Book  II. 

Let  us  suppose  the  origin  of  the  co-ordinates  to  be  at  the  centre 

B 


of  force  C,  and  let  the  line,  CA,  which  joins  the  point  in  which 
the  body  is  at  any  given  time,  to  the  centre,  represent  the  central 
force.  It  will  be  obvious  that  the  three  sides  of  the  parallelepiped 
which  represent  the  three  rectangular  components  of  CA,  will  be 
the  co-ordinates  of  the  point  A,  and  for  any  other  magnitude  of  the 
central  force,  they  will  be  proportioned  to  these  components ; 
hence, 

X  :  T  :  Z  :  :  x  :  y  :  z, 
and 

*Y=7/X  ,  z  X=*Z  ,  y  Z=zY 
xY— yX=0  ,  zX— a:Z=0  ,  yZ-—zY=0, 

and  in  the  former  equations  the  second  members  become  =0 ; 
integrating  these,  we  have 

xdy — ydx=cdt 

zdx — xdx=c'dt 

ydz — zdy=c"dt. 

If  these  expressions  be  multiplied,  the  first  by  2,  the  second  by 
t/,  the  third  by  ar,  they  may  be  added,  and  we  have 

cz+c'y+c"z=0, 

which  is  the  equation  of  a  plane  surface  ;  whence  we  may  infer 
that  the  orbit  is  a  plane  curve.  Take  now  the  projection  of  the 
point  A,  on  the  plane  of  #,  y ;  let  A'  be  this  projection ;  the  radius 
vector,  C  A'=r ;  and  i?,  the  angle  that  determines  its  direction,  in 
respect  to  the  axis  parallel  to  x ;  by  (18) 

x=r  cos.  v,  y—r  sin.  0, 
whence 

xdy — ydx=r2dv  ; 

and  as  the  first  number  of  this  equation  is,  as  we  have  seen,  a  con- 
stant quantity,  so  also  will  be  the  second.  But  we  may  suppose 


Book  //.]  OF  MOTION.  69 

the  arc  of  the  curve,  described  by  the  body  in  passing  between 
the  two  successive  positions,  marked  out  by  the  small  angle  du, 
to  be  a  circle ;  and  the  area  described  by  the  radius  vector  as  a  cir- 

o  _j 

cular  sector  whose  area  is  — - ,  and  this  being  the  half  of  a  con- 
stant quantity,  is  itself  a  constant  quantity  ;  ij;  will  also  be  equal  to 
'  and  consequently,  the  area  described  in  t  is  — ,  wherefore 

the  area  described  during  a  given  time,  is  proportioned  to  the  times. 
The  areas  then,  that  are  described  by  the  radius  vector  of  the  pro- 
jection of  a  moving  point,  upon  a  plane  passing  through  the  centre 
offeree,  are  proportioned  to  the  times.  Now  as  the  orbit  of  the 
body  is  itself  a  plane  curve,  and  as  the  direction  of  the  plane  of 
projection  is  arbitrary,  this  proposition  is  equally  true  in  respect 
to  the  areas  described  by  the  radius  vector  of  the  body  itself,  upon 
the  surface  of  the  plane  curve  which  forms  the  orbit. 

This  proposition  is  the  same  which  has  been  investigated  in 
§  63.  And  is  not  only  true  of  circular  orbits,  but  of  any  orbits 
whatsoever. 

68.  This  same  proposition  may  be  deduced  from  more  simple 
considerations.  In  the  first  place,  the  curve  described  under  the 
action  of  a  central  force,  must  be  a  plane ;  for,  the  direction  of  the 
motion,  in  the  first  element  of  the  time,  will  be  in  the  plane  formed 
by  its  two  components,  the  projectile,  and  the  central  force  ;  the 
direction  in  which  the  moving  point  would  tend  to  go  on,  were 
the  central  force  to  cease  to  act,  will  likewise  be  in  the  same  plane, 
and  the  line,  which  joins  the  extremity  of  this  tangential  force 
to  the  centre,  will  be  in  this  plane  also  ;  this  line  represents  the 
direction  of  the  central  force,  during  the  second  element  of  the 
time,  and  the  resultant  is  therefore  still  found  in  the  original 
plane  ;  and  so  of  all  the  successive  resultants,  which  make  up  the 
curve,  which  lies  therefore  in  one  plane. 

We  have  seen  from  §  33,  that  the  sum  of  the  Moments  of  Ro- 
tation of  any  number  offerees,  is  equal  to  the  Moment  of  Rota- 
tion of  their  resultant.  Now  if  we  suppose  the  curve  to  be  a 
polygon  of  an  infinite  number  of  sides,  and  the  velocities  in  each 
of  these  to  be  uniform,  the  spaces  described  will  be  the  measure 
of  the  several  forces  ;  the  moment  of  rotation  of  the  projectile 
force  will  be  the  tangential  line  4fy  which  represents  that  force, 


Of    MOTION. 


[Book  H. 


multiplied  by  the  radius  vector  CA,  or  twice  the  triangle  ABC  ;  the 
moment  of  rotation  of  the  force  that  acts  in  the 
curve,  is  the  rectangle  under  AC,  and  AD,  or 
twice  the  sectoral  space,  ADC.  And  as  the  di- 
rection of  the  central  force  is  in  the  radius 
vector,  its  moment  of  rotation  =0  ;  hence  the 
twomoments  of  rotation,  of  the  tangential  force, 
and  of  the  force  in  the  curve,  are  equal,  as  are  al- 
so their  respective  halves.  At  the  point  D,  the 
tangential  force  is  that  with  which  the  curve 
has  been  described  ;  its  moment  of  rotation  is 
therefore  equal  to  twice  the  space  DCE,  and 
for  the  same  reason,  it  is  equal  to  twice  the 
second  sectoral  space,  ADC,  which  is  therefore  equal  to  the  sec- 
toral space,  ADF  ;  and  these  are  the  spaces  described  by  the  ra- 
dius vector,  in  the  equal  elements  of  the  time;  hence  the  greater 
spaces  which  these  make  up,  will  be  proportioned  to  the  times. 


Book  //.]  o»  MOTION.  tl 


CHAPTER  VI. 

PRINCIPLE  OF  D'ALEMBERT. 

69.  The  formulae  and  conditions  of  the  motion  and  equilibrium 
of  points,  may  be  applied  to  systems  of  material  points,  or  to  such 
bodies  as  actually  exist  in  nature,  by  means  of  a  self-evident  prin- 
ciple, first  announced  by  D'Alembert,  and  which  has  been  em- 
ployed by  all  succeeding  writers  on  Mechanics  of  any  well 
founded  reputation.  It  may  be  expressed  as  follows  : 

If  there  be  a  body,  or  system  of  points  materially  connected  in 
any  manner  with  each  other,  and  which  are  acted  upon  by  forces 
given  in  magnitude  and  direction  ;  the  action  of  these  several 
forces  is  modified  by  the  connexion  among  the  several  points, 
and  they  neither  move  in  the  direction,  nor  with  the  velocity 
they  would  have,  were  they  not  connected.  Still  the  forces  that 
must  be  compounded  with  those  that  cause  the  motion,  in  order 
to  make  up  the  forces  with  which  the  points  actually  move,  must 
be  such  as  are  in  equilibrio  with  each  other,  or  that  if  they  acted 
upon  the  system  alone,  would  produce  no  motion.  The  last 
mentioned  forces  obviously  represent  the  mutual  action  of  the 
points  upon  each  other  ;  these  could  not  of  themselves  cause  mo- 
tion, and  are  therefore  in  equilibrio. 

To  express  this  analytically,  let  the  forces  applied  to  the  points 
be  represented  by  the  velocities  a,  6,  c,  &c. ;  the  velocities  the 
points  actually  have,  by  a  b\  c  ;  the  velocities  which  must  be 
combined  with  the  first  of  these,  in  order  to  produce  the  last,  a'', 
6",  c",  &c.,  then 

a"-f-fc"+c"-f&c.=:0; 

for  it  is  obvious  that  these  velocities  have  no  effect  upon  the  act- 
ual moving  force  of  the  system,  which  is  due  to  the  velocities  a, 
6,  c,  &c.,  alone  ;  and  hence  the  forces  which  they  represent  mu- 
tually destroy  each  other. 

To  apply  this  to  systems  of  bodies,  we  must  consider  that  the 
moving  force  of  each  will  depend  not  only  on  its  velocity,  but  on 
the  number  of  equal  material  points  it  contains ;  therefore,  the 
force,  by  which  each  body  is  impressed,  is  due  to  the  product  of 
its  number  of  points  into  its  velocity;  calling  the  former,  A,  B, 
C,  &c.,  we  have 

Aa"-r-B&"4-Cc"-|-&c.=0.  (98) 


72  OF  MOTION.  [Book  II. 

CHAPTER  VII. 

PRINCIPLE  OF  VIRTUAL  VELOCITIES. 

70.  If  we  suppose  the  equilibrium   of  any  system   of  forces 
whatsoever  to  be  disturbed  for  an  instant,  and  that  each  point  to 
which  a  force  is  applied,  has  a  velocity,  such  as  would  have  been 
given  to  it,  in  the  direction  of  the  disturbance,  by  the  force  which 
acts  upon  it.     The  velocities  that  the  several  points  would  ac- 
quire, are  called  their  Virtual  Velocities. 

71.  In  any  system  of  forces  whatsoever,  the  sum  of  the  pro- 
ducts of  the  several   forces,  into  their  respective  virtual  veloci- 
ties, is  equal  to  0,  or  calling  the  forces,  F,  F',  F",  the  virtual  ve- 
locities, v,  v',  v", 

Ft>+FV+FV+&c.=0.  (99) 

Forces  can  only  act,  under  certain  definite  circumstances,  upon 
their  points  of  application. 

1.  All  the  forces  may  he  applied  to  a  single  free  point. 

2.  The  point  of  application  of  the  forces  may  be  compelled  to 
rest  upon  a  given  surface. 

3.  It  may  be  compelled,  when  it  does  move,  to  move  along  a 
given  surface. 

4.  The  forces  may  act  upon  a  system  of  points,  united  toge- 
ther in  any  possible  manner. 

5.  The  system  may  have  a  fixed  point,  around  which  the  others 
must  move,  or  a  point  that  is  compelled  to  describe  a  given  sur- 
face. 

First.  Let  the  forces  act  upon  a  single  point,  C,  in  the  figure 
beneath;  let  FC,  FC',  FC",  &c.,  be  their  several  directions 
and  RC,  the  .direction  of  their  resultant  R. 


Draw  a  line,  Cd,  to  represent  the  direction  of  the  disturbance,  or 
that  in  which  the  point  is  supposed  to  move,  and  the  distance  to 


Book  IL]  OF  MOTION.  73 

which  it  is  removed.  This  line  will  therefore  represent  the  vir- 
tual velocity  of  this  point.  Decompose  each  of  the  forces  into 
two  others,  one  parallel  to  the  line  Cd,  the  other  perpendicular  to 
it;  the  respective  values  of  the  components  vill  be,  calling  the 
angles  that  they  respectively  make  with  Cd,  a'  and  6,  6',  6",  &c., 

R  cos.  #,  F  cos.  6,  F  cos.  6',  F  cos.  6",  &c. 

Now  if  perpendiculars  be  let  fall  from  the  point  d,  upon  the  several 
directions  of  these  forces,  the  angles  these  will  respectively  make 
with  the  line  Cd,  will  be  equal  to  the  angles  the  directions  of  the 
forces  make  with  it  at  the  point  C.  The  value  of  the  projections, 
r, /,/,/",  &c.,  will  therefore  be  symmetric  with  the  values  of 
the  components  of  the  forces  parallel  to  Cd,  or  callingthe  lineCd,  c, 

r=c  cos.  a,  f=c  cos.  6,  f'—c  cos.  6',  &c. 

Now,  as  the  sum  of  the  components  of  any  force,  estimated  in  a 
given  direction,  is  by  (19)  equal  to  the  component  of  the  force 
estimated  in  the  same  direction, 

R  cos.  a=F  cos.  6-f-F'  cos.  &'-j-F"  cos.  6"+&c. 

multiplying  this  by  the  value  of  c,  obtained  from  the  foregoing  ex- 
pressions, we  obtain 

Rr=F/+F'/+F'/"+&c. 

but  r  is  the  virtual  velocity  of  the  point  C,  and/,/',/",  &c.,  are 
obviously  the  components  of  that  velocity,  in  the  several  directions 
of  the  component  forces,  and  are  therefore  so  much  of  the  vir- 
tual velocity  as  is  due  to  the  action  of  the  respective  forces  ;  sub- 
stituting then  the  letters  u,  i/,  u",  &c.  for/,/,/',  &c.,  and  sup- 
posing the  system  to  be  in  equilibrio,  we  obtain 

F«+FV+FV'+&c.=0. 

Second.  In  the  case  of  the  system  of  forces  being  such  as 
causes  a  point  to  rest  upon  a  given  surface,  the  action  of  this  sur- 
face may,  §  19,  be  represented,  by  introducing  a  force  normal  to 
the  surface  in  its  direction,  and  equal  to  the  resultant  of  the  other 
forces,  which  is  also  a  normal  to  the  surface.  This  force  may 
therefore  be  introduced  among  the  forces  in  the  above  equation ; 
calling  it  P,  and  the  virtual  velocity  it  causes,  j>,  we  have 

Pp-hF/fF/-r-F'/''-f&c.==0; 
,     and 

F/-f-F/-j-F"/'+&c.=0 ; 

the  proposition  is  therefore  true  in  respect  to  points  compelled  to 
rest  upon  a  surface  by  the  action  of  the  forces. 

Third.  The  same  reasoning  applies  to  the  case  where  the  point 
is  compelled,  if  it  move,  to  move  along  a  given  curve ;  the  mo- 
tion being  very  small,  may  be  considered  as  taking  place  in  the 
direction  of  the  tangent  to  the  curve,  or  perpendicular  to  the  di- 
10 


74  09  MOTION.  [Book  If. 

rection  of  P  ;  the  virtual  velocity  of  p  is  therefore  equal  to  0,  for 
the  angle  of  inclination  becomes  90'  ;  and 

P/>=0; 
hence  again, 

F/+F'/-fF"/"+&c.=0. 

^- 

Fourth.  If  the  forces  act  upon  points  composing  a  system,  in 
which  they  are  united  in  any  manner  whatsoever  ;  call  the  forces, 
which  we  shall  for  the  present  restrict  to  three  in  number,  F,  F', 
F".  Each  point  is  held  in  its  position,  in  the  case  of  equilibrium, 
by  actions  exerted  in  consequence  of  its  connexion  with  the  other 
points  in  the  system  ;  these  actions  may  be  represented  by  forces, 
coinciding,  in  their  several  directions,  with  the  lines  that  join  each 
pointto  the  others  which  compose  the  system,  and  these  forces  will 
have  a  determinate  magnitude.  Call  the  forces,  that  represent 
these  mutual  actions,  on  the  three  several  points,  A,  A',  B,  B', 
C,  C'.  Now  as  each  point  will  be  in  equilibrio  under  the  action 
of  three  forces,  that  which  is  applied  to  it,  and  the  two  which  are 
exerted  upon  it  by  the  others,  the  principle  of  virtual  velocities 
holds  good  in  respect  to  each  of  these  three  sets  offerees,  or 


F'/"+Cc-f  CV=0  ; 
but  the  points  being  mutually  connected, 

A'=B,  B'=C,  C'=A  ; 
hence 

F/+Aa+Ba'=0, 


"/'+  Cc+Ac'=0  ; 
adding  these  equations, 


But  the  equilibrium  of  the  system  would  subsist,  if  the  forces  F, 
F',  F",  were  suppressed,  and  the  system  supported  by  forces 
equal  and  opposite  to  A,  A',  B,  B',  C,  C',  applied  to  the  several 
points  ;  for  each  of  these  pairs  of  forces  has  for  its  resultant  a 
force  equal  and  opposite  to  F,  F',  and  F"  respectively.  The 
principle  of  virtual  velocities  is  therefore  applicable  to  this  new 
set  of  forces  also  ;  or 


—  A(a+c')—  B 
subtracting  this  equation  from  the  former,  we  have 


And  although  our  reasoning  has,  in  order  to  prevent  complexity, 
been  restricted  to  three  points  and  three  forces  ;  it  is  obvious 
that  it  might  have  been  extended  to  any  number  whatsoever. 


//.]  op  MOTION.  75 

As  we  have  assumed  the  points  to  be  connected  in  any  manner 
whatsoever,  it  is  obvious  that  the  principle  is  applicable  to  all  cases 
of  free  systems  of  material  points,  whether  they  be  united  by  in- 
variable and  inflexible  lines,  or  by  the  loose  aggregation  that  takes 
place  in  the  particles  of  fluids,  or  by  any  intermediate  connexion 
between  that  which  is  fixed  and  invariable,  and  that  in  which  the 
connexion  is  about  to  cease  altogether. 

Fifth.  In  the  same  manner,  precisely,  in  which  the  case  of 
forces,  acting  upon  a  single  free  point,  has  been  applied  to  those 
of  forces  acting  upon  a  point  compelled  to  rest  or  to  move  upon  a 
given  surface,  may  the  above  case  be  extended  to  those  of  systems 
that  have  in  them  a  fixed  point,  or  a  point  compelled  to  move 
upon  a  given  surface,  and  thus  the  principle  of  Virtual  Velocities 
may  be  shown  to  be  true  in  all  possible  cases. 


BOOK  III. 


OF  THE  EQUILIBRIUM  OF  SOL.ID  BODIES, 


CHAPTER  I. 

GENERAL  PROPERTIES  OF  MATTER.   DIVISION  OF  NATURAL  BODIES. 
MEASURE  OF  THE  MOVING  FORCES  OF  BODIES. 

72.  In  order  to  apply  the  preceding  principles  to  the  circum- 
stances which  occur  in  nature,  it  becomes  necessary,  that  we 
should  investigate  not  only  the  general  abstract  properties  of  mat- 
ter, but  also  those  which  we  find  inherent  in  the  greater  part,  if 
not  the  whole,  of  these  material  substances  that  we  can  make  the 
objects  either  of  experiment  or  observation. 

These  essential  properties  of  matter,  without  which  we  cannot 
conceive  it  to  exist,  are,  Extension,  Mobility,  and  Impenetrability. 

73.  Matter  must  occupy  a  certain  portion  of  space,  and  thus  be 
extended  in  three  dimensions,  or  is  of  that  class  of  Geometric 
magnitude,  which  is  called  a  Solid. 

74.  Beingextended,  it  is  of  course  capable  of  division,  and  were 
we  to  reason  in  respect  to  this  secondary  property,  as  if  the  ma- 
terial substance  had  the  same  geometric  properties  with  the  space 
it  occupies,  it  might  be  demonstrated  that  matter  is  infinitely  di- 
visible ;   and  thus,  in  many  works  on  Mechanical  Philosophy,  it 
has  been  inferred,  from  strict  mathematical  reasoning,  that  this  is 
the  fact.     It  is  however,  obvious,  that  the  reasoning  which  is  ap- 
plicable to  mere  geometric  magnitude,  has  reference  only  to  the 
space  occupied  by  matter,  and  not  to  matter  itself.  We  may,  there- 
fore, reasonably  doubt  whether   matter  be  infinitely  divisible. 

Still,  the  divisibility  of  matter  may  be  carried  to  an  extent,  that 
exceeds  the  limits  that  our  senses  can  reach  :  thus  by  mere  me- 
chanical division,  as  in  the  hammering  and  drawing  of  metals, 
the  divisibility  of  the  substances  is  very  remarkable  ;  gold,  when 
manufactured  into  leaf,  is  beaten  out,  until  the  thickness  of  the 
sheet  does  not  exceed  anVurth  Part  °f  an  inch. 


7S  Of    THE    EQUILIBRIUM  [Book  III. 

In  the  mixed  mechanical  and  chemical  arts,  divisibility  is  car- 
ried still  farther ;  in  making  gold  lace,  for  instance,  where  a  rod 
of  silver  is  gilt  by  means  of  an  amalgam,  and  then  drawn  into  wire, 
a  single  grain  of  gold  covers  9600  square  inches ;  and  when  the 
wire  is  flattened,  the  surface  covered  is  nearly  doubled  in  extent. 

Chemistry  furnishes  still  more  marked  instances  of  divisibility ; 
particularly  in  the  manner  in  which  the  colours  produced  by  che- 
mical tests  and  re-agents,  are  diffused  throughout  spaces  of  con- 
siderable magnitude,  by  quantities  of  substances  that  are  hardly 
appreciable  : 

There  are  also  substances,  that  we  know  by  our  senses  to  exist, 
or  can  distinguish  by  the  effects  they  are  capable  of  producing, 
which  escape  the  nicest  methods  of  chemical  research  ;  thus,  the 
matter  of  which  many  of  the  most  powerful  odours  is  composed, 
has  not  been  detected  by  any  analysis  of  the  air  which  is  its  ve- 
hicle ;  and  the  pestilential  miasmata  which  cause  disease,  and 
even  certain  death,  have  baffled  the  most  powerful,  as  well  as 
the  most  delicate  methods  of  inquiry  : 

Another  proof  of  the  great  divisibility  of  matter,  may  be  drawn 
from  the  minuteness  of  the  living  animals  that  can  be  distinguished 
by  the  aid  of  the  microscope  ;  Leuwenhoeck  saw  animals  not  ex- 
ceeding the  ten  thousandth  part  of  an  inch  in  length,  and  each  of 
these  is  furnished  with  organs,  has  vessels  in  which  fluids  circu- 
late, whose  particles  therefore  must  be  small  beyond  all  con- 
ception. 

Mere  physical  action  is  also  capable  of  exhibiting  a  very  exten- 
sive divisibility,  in  the  matter  that  is  subject  to  it ;  thus  water, 
when  heated  beyond  a  certain  temperature  is  converted  into  steam, 
which  occupies  about  seventeen  hundred  times  as  much  space  as 
the  water  whence  it  is  generated,  and  yet  no  perceptible  portion 
of  the  space  is  devoid  of  aqueous  matter. 

75.  The  term  Body,  is  employed  to  denote  a  determinate 
quantity  of  matter,  contained  under  some  known  figure,  or  exist- 
ing in  some  peculiar  mode.  By  mechanical,  physical,  or  chemi- 
cal action,  influencing  the  divisibility  of  matter,  a  body  may  be 
disintegrated  ;  its  component  parts  wilt  assume  new  forms  and 
characters,  and  constitute  new  bodies.  By  such  forces  constantly 
acting  in  nature,  the  surface  of  the  globe,  and  all  the  bodies  that 
we  find  upon  it,  are  undergoing  continual  changes  ;  and  are  con- 
stantly assuming  new  forms,  or  entering  into  new  combinations. 
In  all  these  changes,  however,  no  portion  of  matter  is  annihilated, 
but  its  quantity  continues  invariable.  We  know  in  fact  of  no 
agent  in  nature,  that  is  capable  of  increasing  or  diminishing  the 
amount  of  the  matter  that  exists  in  the  Universe.  Matter,  there- 
fore, might  be  inferred  to  be  eternal  ;  but  as  we  find  that  it  is 


Book  III.}  OF  SOLID  BODIES*  79 

not  only  a  metaphysical  truth,  but  one  that  the  process  of  induc- 
tion enables  us  to  deduce,  from  a  full  investigation  of  all  the  phe- 
nomena of  nature  :  that,  nothing  can  exist  in  any  state  whatever, 
without  some  sufficient  reason  ;  we  have  a  right  to  infer,  that  the 
permanence  of  matter,  is  the  result  of  the  action  of  a  great  and 
all  powerful  first  cause. 

76.  The  indestructibility  of  matter  might  be  urged  as  a  proof 
that  it  cannot  be  infinitely  divisible,  for  that  property  could  only 
result  in  its  annihilation.  But  we  are  enabled  to  draw  more  con- 
clusive evidence^of  the  fact,  that  the  divisibility  of  matter  is  finite, 
from  the  modern  discoveries  of  chemistry.     In  this  science,  we 
are  enabled  to   discover,  and  explain  many  important  facts,  by 
means  of  the  hypothesis,  called  the  Atomic  Theory ;  this  holds, 
that  all  bodies  are  finally  resolvable  into  particles,  incapable  of 
farther  division.     From  what  has  been  said,  it  will  however  ap- 
pear, that  these  atoms  must  be  so  small  as  to  escape  the  mdst 
acute  of  our  senses,  even  when  furnished  with  the  most  pow^r- 
ful  aids  that  the  high  improvement  of  the  arts  at  present  affords. 
The  proof  of  the  existence  of  atoms,  therefore,  is  not,  and  proba- 
bly cannot  be,  complete  ;    for  to  constitute  a  theory  that  can  be 
received  as  absolutely  true  in  philosophy,  it  is  not  only  necessary 
that  it  should  fully  explain  all  the  phenomena,  but  that  it  shoild 
be  founded  on  evidence  entirely  independent  of  them.     Stilljas 
in  the  most  minute  state  of  division,  to  which  bodies  can  be  De- 
duced in  practice,  we  find  them  retaining  their  peculiar  and  infli- 
vidual  properties,  we  have  a  right  to  infer,  that  their  divisibility 
is  in  no  case  carried  to  an  infinite  extent ;  we  may,  therefore, 
assume,  that  they  are  made  up  of  portions  infinitely  small ;  th^se 
are  usually  called  Particles. 

77.  The  term  Impenetrability,  as  applied  to  denote  a  property 
of  matter,  merely  implies,  that  no  two  particles  can  occupy  ihe 
same  space,  at  the  same  time.    In  this  point  of  view,  the  hardest 
and  the  softest  substances  are  equally  impenetrable.     Impenetra- 
bility is  absolutely  essential  to  the  existence  of  matter,  and  has 
an  experimental  proof  in  its  indestructibility;  for,  were  it  not 
impenetrable,  two   portions   of    matter  might  be  reduced  to  a 
single  one,  and  one  of  them  annihilated,  which  we  have  seen  to 
be  impossible.     It  is  to  this  properly,  of  excluding  all  other  por- 
tions of  matter  from  the  space  that  themselves  occupy,  that  we 
have  recourse,  in  ascertaining  the  material  existence  of  certain 
substances,  concerning  which,  inaccurate  notions  long  prevailed. 

78.  The  property  of  mobility  is  also  deduced  from  experiment 
and  observation,  for  we  find  no  body  that  cannot  be  set  in  mo- 
tion, by  an  adequate  force,  nor  any  whose  motion  cannot  in  like 


80  OP    THE    EQUILIBRIUM  [Book  III. 

manner,  be  arrested.  It  might  also  be  shown  to  be  a  conse- 
quence of  the  Impenetrability  of  matter,  for  as  a  body  resists  the 
entrance  of  any  other,  into  the  space  itself  occupies,  motion  must 
arise  whenever  a  body,  impelled  by  any  force  whatever,  tends 
to  enter  the  space  another  previously  occupies. 

The  sum  of  all  the  particles  in  a  body,  constitutes  its  mass,  or 
makes  up  its  quantity  of  matter  ;  and  in  homogeneous  bodies,  the 
quantity  of  matter  is  obviously  proportioned  to  their  respective 
bulks. 

In  heterogeneous  bodies,  the  quantity  of  matter  has  respect 
not  only  to  the  bulk,  but  to  the  Density;  the  latter  is  the  rela- 
tion that  the  quantity  of  matter,  in  a  given  body,  bears  to  its  bulk. 

This  definition  may  be  thus  expressed  : 

D=^;  (100) 

whence  we  have 

M 
M=DB,  andB=jj;  (101) 

The  Densities  then,  are  directly  as  the  quantities  of  matter, 
and  inversely  as  the  Bulks  : 

The  Quantities  of  matter,  are  in  the  compound  ratio  of  the 
Densities  and  Bulks;  and 

The  Bulks,  are  directly  as  the  quantities  of  matter,  and  inversely 
as  ihe  Densities. 

79.  Bodies  differ  from  each  other,  in  the  greater  or  less  diffi- 
culty with  which  their  particles  may  be  separated.     When  the 
separation  of  the  particles  requires  the  application  of  a  determinate 
force,  the  body  is  said  to  be  solid  ;   when  they  may  be  divided,  by 
a  force  so  small  as  to  be  inappreciable,  they  are  said  to  be  fluid.    Of 
flu  ds,  some  have  so  small  a  capability  of  having  their  bulk  changed 
by  pressure,  as  to  have  been  considered  as  absolutely  incompres- 
sible ;   such  bodies  are  called  Liquids  ;  of  them,  water  at  ordinary 
temperatures  is  a  familiar  instance.     Others  again,  are  capable  of 
being  compressed,  and  of  occupy  ing  larger  spaces,  when  the  com- 
pressing force  is  removed  ;    these  are  styled  Gases,   or  Elastic 
Fluids  ;  of  these  atmospheric  air,  may  be  cited  as  the  type. 

80.  Many  bodies  are  familiarly  known  to  be  capable,  under 
different  physical  circumstances,  of  existing  in  all  the  three  dif- 
ferent states  :  thus,  water,  when  cooled  below  a  certain  tempera- 
ture, passes  into  the  solid  form,  and  becomes  Ice  ;  while  if  heat- 
ed, beyond  a  certain  limit,  it  assumes  the  gaseous  state,  and  is 
called  Steam.     As  a  general  rule,  deduced  from  various  chemical 
and  purely  physical  facts,  every  body  in  nature  is  capable  of  assu- 


///.]  OF    SOLID    BODIES.  81 

ming,  under  proper  circumstances,  all  of  these  three  mechanical 
states. 

81.  Heat  is  the  great  natural  agent  that  is  concerned  in  these 
mechanical  changes,  and  it  is  a  general  rule,  that  heat,  existing  in 
that  modification  in  which  it  is  said  to  be  latent,  determines  the 
mechanical  state  that  bodies  assume. 

When  the  latent  heat  is  withdrawn,  the  body  returns  to  its 
original  state  ;  thus  steam,  on  parting  with  its  latent  heat,  be- 
comes water  ;  and  water,  on  parting  with  its  latent  heat,  becomes 
ice  :  hence,  we  infer  the  action  of  another  force  in  opposition  to 
that  of  heat;  to  this  force,  whose  cause  is,  like  that  of  heat,  un- 
known to  us,  we  give  the  name  of  the  Attraction  of  Aggregation. 
These  two  great  natural  antagonist  forces  then,  determine  the 
mechanical  states  in  which  bodies  exist. 

When  the  attraction  of  aggregation  predominates,  the  body  is 
a  Solid. 

When  they  are  equally  balanced,  the  body  is  a  perfect  Liquid ; 
and  when  the  force  of  heat,  exceeds  that  of  the  attraction  of  ag- 
gregation, the  body  becomes  an  Elastic  Fluid. 

It  will  be  hereafter  seen,  that  no  body  has  a  constitution  that 
will  entitle  it  to  the  name  of  a  perfect  liquid.  In  all  known  bo- 
dies of  this  class,  the  force  of  attraction  still  preponderates  over 
that  of  heat,  as  will  be  made  manifest  when  we  examine  the  forms 
in  which  small  masses  of  liquids  arrange  themselves. 

82.  The  attraction  of  aggregation  is  only  known  to  us  as  act- 
ing at  insensible  distances ;  it  is  therefore  unnecessary  to  enter 
into  any  investigation  of  the  ratio,  in  which  its  intensity  dimi- 
nishes, as  the  distance  increases.     Its  absolute  measure  differs  in 
every  different  body,  and  must  hence  be  determined  experimen- 
tally ;  it  constitutes  the  strength  of  the  materials  that  are  em- 
ployed in  practical  mechanics,  and  will,  under  this  name,  become 
the  object  of  future  consideration.     Although  the  general  princi- 
ples of  the  two  first  books,  apply  equally  to  all  bodies,  whatever  be 
their  state  of  aggregation  ;  still,  they  are  modified  in  their  action 
by  the  peculiar  mechanical  properties  of  the  different  classes  into 
which  we  have  found  them  to  be  divided.     It  is  hence  necessary 
to  consider  the  Mechanics  of  Solid,  and  of  Fluid  Bodies,  sepa^ 
rately. 

83.  In  our  previous  inquiries,  forces  have  been  considered  in 
the  abstract,  and  as  acting  upon  material  points  of  indeterminate 
magnitude,  although  we  have  occasionally  been  compelled  to  use 
the  term  Body.     It  ROW  becomes  necessary  that  we  should  be 
able  to  estimate  the  force  by  which  a  body  or  aggregate  of  matter 
is  actuated.     This,  which  is  called  the  Quantity  of  Motion,  may 

11 


82  OF   THE    EQUILIBRIUM  [Book  III. 

be  ascertained  from  the  following  considerations.  We  may  ob- 
viously consider  a  moving  body  as  made  up  of  its  particles,  or  of 
a  number  of  material  points,  and  supposing  its  motion  to  be  recti- 
lineal, every  particle  will  be  actuated  by  a  force,  equal  and  paral- 
lel to  those  which  actuate  the  rest  ;  the  whole  force  then  will  be 
equal  to  the  mass  of  the  body,  multiplied  by  the  forces  that  actu- 
ate its  particles.  If  the  motion  be  uniform,  the  velocity  of  each 
of  these  particles  is  constant,  and  will  represent  the  force  by 
which  it  is  produced  ;  hence,  the  moving  force  of  a  body  will  be 
represented  by  its  mass,  multiplied  by  its  velocity.  The  same 
will  be  obvious  from  other  considerations  ;  for  it  is  clear,  that 
when  several  bodies  have  equal  velocities,  that  which  has  twice 
the  quantity  of  matter  that  another  has,  must  have  twice  the  quan- 
tity of  motion,  and  so  on.  In  like  manner,  if  the  masses  be  equal, 
the  body  which  has  the  greatest  velocity  will  have  a  quantity  of 
motion  exactly  proportioned  to  it. 

This  may  be  illustrated  thus  : 

Let  M  be  the  mass  of  a  body  made  up  of  the  several  particles 
p,  p'i  p",  &c.,  and  v  the  common  velocity,  the  sum  of  the  several 
forces  by  which  they  are  actuated  will  be 

^v  +  &c.  , 


or   pppc.     Xv  ; 

and  as  the  sum  of  the  particles  is  equal  to  the  mass,  the  expres- 
sion for  the  quantity  of  motion,/,  wiU  become 

/=Mr;  (102) 

whence 

«=^-  (103) 

From  these  equations  the  following  consequences  immediately 
follow  : 

(1).  The  forces  of  moving  bodies,  or  their^quantities  of  motion, 
are  represented  by  the  products  of  their  masses  and  velocities. 

(2).  In  equal  masses,  the  forces  are  proportioned  to  the  velo- 
cities. 

(3).  When  bodies  have  equal  quantities  of  motion,  the  velo- 
cities are  inversely  as  their  masses. 

(4).  With  equal  velocities,  the  quantities  of  motion  are  pro- 
portioned to  the  masses. 


Book  ///.]  OF    SOLID    BODIES,  83 

CHAPTER  II. 

ATTRACTION  OF  GRAVITATION. 

84.  It  is  a  fact  demonstrated  by  universal  experience,  that  all 
heavy  bodies,  if  unsupported,  fall  towards  the  surface  of  the  earth. 
This  can  only  take  place  in  consequence  of  the  action  of  some 
specific  force,  which  has  been  supposed  to  reside  in  the  earth  it- 
self, and  which  has  been  called  the  Attraction  of  Gravitation,  or 
more  simply,  Gravity. 

85.  The  direction  of  this  force  may  be  ascertained  by  suspend- 
ing a  heavy  body,  from  a  fixed  point,  by  a  flexible  string ;  the 
direction  of  the  string  will,  as  is  obvious,  mark  the  line  in  which  a 
body  would  tend  to  move,  under  the  action  of  the  force. 

Such  an  apparatus  is  called  the  Plumb-Line.  When  adapted 
to  a  ruler  with  parallel  sides,  it  becomes  a  familiar  and  simple 
instrument,  of  great  practical  utility.  The  plumb-line  hanging 
freely,  being  made  to  coincide  with  a  line  drawn  along  the  ruler 
parallel  to  its  sides,  the  latter  also  point  out  the  direction  of  gravity. 

If  a  second  ruler  be  adapted  to  the  lower  extremity  of  the  first, 
and  make  with  it  angles  that  are  exactly  90°,  the  face  of  the 
second  ruler  will  D*e  found  to  adapt  itself  to  the  surface  of  stag- 
nant waters;  hence  we  infer  that  the  direction  of  gravity-is  per- 
pendicular to  the  surface  of  standing  water.  We  say  however, 
in  general  terms,  that  the  direction  of  gravity  is  perpendicular 
to  the  surface  of  the  earth. 

We  are,  in  truth,  compelled  to  refer  the  direction  of  gravity  to 
some  more  extended  surface  than  that  of  the  largest  mass  of 
tranquil  water  :  for  it  is  a  question  that  requires  examination, 
whether  the  direction  of  gravity,  at  a  given  place,  be  constant. 
To  ascertain  this,  it  would  be  necessary  to  have  some  points 
that  we  could  consider  absolutely  fixed.  Even  the  most  stable 
edifice  has  not  sufficient  permanence,  to  afford  points  that  can  be 
relied  upon  as  such.  If,  for  instance,  we  were  to  find,  that  the 
relative  direction  of  the  plumb-line  and  the  wall  of  a  building 
had  changed,  we  might  at  first  sight  doubt  which  had  altered  its 
position  ;  for  we  know,  that  natural  convulsions,  or  even  gra- 
dual decay,  may  alter  the  direction  of  the  firmest  walls.  Even 
the  sides  of  a  mountain  could  not  be  relied  upon,  for  we  know 
of  mountains  being  affected  by  earthquakes. 

The  sea,  however,  apparently  unstable  as  it  at  first  sight  appears, 
has,  in  its  mean  level,  and  in  the  limits  within  which  its  ebb  and 
flow  are  bounded,  an  instance  of  the  greatest  stability  that  we 


84  OF    THE    EQUILIBRIUM  [Book  III. 

find  on  the  face  of  our  globe.  Were  the  mean  height  of  its  sur- 
face to  change,  or  the  limits  of  its  rise  to  be  much  extended,  it 
would  be  attended  with  the  most  disastrous  consequences,  caus- 
ing great  inundations,  or  even  another  deluge.  Were  the  sea  to 
become  still,  and  to  be  no  longer  agitated,  either  by  the  waves 
that  are  raised  by  the  winds,  or  those  which  constitute  its  tides, 
it  would  come  to  rest,  in  a  position  whose  surface  would  be  a 
mean,  between  the  limits  within  which  its  oscillations  now  take 
place.  This  surface  it  will  hereafter  be  shown,  must  be  every 
where  perpendicular  to  the  direction  of  gravity  ;  for  the  present, 
we  must  receive  this  fact  as  true,  from  the  experimental  illustra- 
tion that  has  been  cited.  It  is  this  mean  surface  of  the  ocean, 
supposed  to  be  produced  through  the  body  of  the  continents,  and 
other  portions  of  dry  land,  that  we  understand  by  the  Surface  of 
the  Earth. 

86.  The  earth  being  a  body  nearly  spherical,  as  can  be  shown 
by  a  variety  of  astronomic  facts,  as  well  as  from  actual  observa- 
tion and  measurement,    all  the  directions  of  gravity  converge 
towards  its  centre.     They  would  actually  meet  there,  were  the 
earth  a  perfect  sphere. 

87.  Ordinary  observation  might  lead  us  at  first  sight  to  sup- 
pose, that  bodies  are  very  unequally  influenced  by  the  attraction 
of  gravitation  ;  that  its  action  had  relation  not  only  to  their  masses, 
but  was  influenced  also  by  their  densities.     Thus  the  metals  and 
stones  fall  with  great  velocity ;  wood  and  other  vegetable  sub- 
stances less  rapidly  ;  while  feathers,  down,  and  other  similar  sub- 
stances, seem  hardly  to  be  affected   by  the  earth's  attraction. 
Others  again  actually  rise  from  the  surface,  apparently  in  oppo- 
sition to  the  direction  of  the  attractive  force,  as  vapour,  smoke, 
and  clouds.    When,  however,  we  consider,  that  some  bodies  which 
we   may  under  ordinary  circumstances  see  to  fall,   will,  under 
others,  rise  upwards,  we  are  tempted  to  examine  whether  analo- 
gous causes  may  not  exist,  to  determine  the  floatation,  or  actual 
elevation  of  others.     Thus,  for  instance,  a  piece  of  wood  that  falls 
in  the  open  air,  rises  when  placed  in  water,  and  floats  at  the  sur- 
face; and  even  iron  will  do  the  same,  when  plunged  in  a  mass  of 
mercury.    The  air  then,  it  is  possible,  may,  by  its  buoyancy,  pre- 
vent altogether  the  fall  of  some  bodies,  and  even  support  them 
in  the  atmosphere  ;  and  may,  by  its  resistance,  retard  the  descent 
of  others.     Whether  this  be  the  fact  or  not,  may  be  tested  by 
means  of  the  air  pump.     This  apparatus,  as  will  hereafter  be 
seen,  is  capable  of  exhausting  the  greatest  part  of  the  air,  from  a 
proper  vessel,  called  a  receiver  ;  and  in  the  exhausted  receiver  of 
an  air-pump,  solid  bodies  of  every  diversity  of  density  fall  in 
times  that  are  absolutely  equal ;  while  the  lightest  vapours,  and 


///.]  OF    SOLID    BODIES.  85 

smoke,  descend  and  occupy  the  bottom  of  the  vessel.  Thus  all 
bodies  are  influenced  by  gravity,  and  the  velocities  of  falling 
bodies  being  thus  shown  to  be  equal,  when  not  resisted  by  the 
«ir,  whatever  be  their  densities,  it  follows,  that  the  forces  which 
actuate  them,  are  proportioned  to  the  masses,  or  quantities  of 
matter. 

88.  As  the  earth  is  situated  in  free  and  open  space,  it  follows 
from  what  has  been  said  in  relation  to  the  inertia  of  matter,  that 
it  cannot  impress  motion  on  a  body,  without  parting  with  an  equal 
quantity  of  its  own  motion  ;  or,  in  simpler  terms,  that  it  must 
move  towards  the  falling  body,  with  a  quantity  of  motion,  equal 
to  that  with  which  the  body  moves  towards  it.     But  the  mass  of 
the  earth  is  so  vast  in  respect  to  the  bodies  which  we  can  perceive 
to  fall,  that  the  velocity  of  the  earth  towards  the  falling  body 
will  be  wholly  inappreciable.     We  cannot  therefore  test  by  ob- 
servation, the  fact,  whether  the  fall  of  a  heavy  body  is  caused  by 
the  mutual  action  of  the  earth  and  the  body,  or  whether  the  body 
alone  moves.     Neither  can  we  admit  mere  reasoning  to  decide 
whether  the  attraction  be  mutual,  or  is  exerted  solely  by  the 
greater  mass,  which  is,  in  this  case,  the  earth. 

89.  If  the  attraction  between  a  falling  body  and  the  whole 
mass  of  the  earth  be  mutual,  it  will  follow,  that  it  must  also  be 
mutual  between  the  parts  that  make  up  the  mass  ;  and  the  at- 
tracting force  of  the  earth  will  be  due  to  the  sum  of  the  separate 
attracting  forces  with  which  its  particles  are  endowed.     A  force 
thus  made  up,  will  be  influenced  by  irregularities  on  the  sur- 
face of  the  earth  ;  and  large  projecting  masses,  such  as  mountains, 
would  cause  a  deflection  in  the  plumb-line,  from  a  perpendicular 
to  the  general  surface  of  the  earth. 

Whether  such  a  deviation  do  occur,  can  only  be  determined 
by  the  aid  of  astronomic  observation.  The  deviation  must  be  at 
most  extremely  small,  for  the  size  of  the  largest  mountains  bears 
but  a  very  small  proportion  to  the  whole  mass  of  the  earth  :  it  is 
therefore  wholly  imperceptible,  except  to  the  most  accurate  modes 
of  observation,  and  it  is  only  in  mountains  of  considerable  size, 
that  it  can  be  detected,  even  by  them.  But  if  such  deviation  of 
the  plumb-line,  from  the  true  vertical  shall  be  detected,  it  furnishes 
full  and  complete  evidence,  that  the  attraction  of  gravitation  is 
mutual  between  the  bodies  that  are  affected  by  it. 

The  deflection  of  the  plumb-line  by  the  attraction  of  a  moun- 
tain, was  first  suspected  by  Bouguer,  one  of  the  French  acade- 
micians who  were  sent  to  Peru  for  the  purpose  of  measuring  an 
arc  of  the  Meridian.  In  carrying  on  a  series  of  triangles,  in  the 
neighbourhood  of  the  great  mountain  of  Chimborazo,  latitudes 
were  determined  by  means  of  observations  of  the  stars,  both  on 


86  OP    THE    EQUILIBRIUM  [Book  HI. 

the  north  and  south  sides  of  the  mountain  ;  the  itinerary  measure 
between  two  of  these  stations,  was  not  found  to  correspond  to  the 
difference  of  latitude,  and  as  the  apparent  latitude  depends  on 
the  position  of  the  zenith,  pointed  out  by  the  plumb-line,  this 
discrepancy  showed  a  deflection  in  the  latter.  The  deflection  ob- 
served by  Bouguer,  did  not  appear  to  exceed  7"  or  8". 

In  1772,  Maskelyne,  the  British  Astronomer  Royal,  performed 
a  series  of  observations  of  the  same  character,  at  the  base  of  the 
mountain  Schehallien,  in  Scotland.  The  result  of  these  was,  to 
show  conclusively,  the  fact  of  the  deviation  of  the  plumb-line ; 
and  from  his  having  this  sole  object  in  view,  and  from  the  accu- 
racy of  his  observations,  we  can  place  the  most  implicit  confi- 
dence in  his  inferences.  The  deviation  was  found  to  amount 
to  54". 

It  will  be  at  once  seen,  that,  if  the  density  and  bulk  of  the 
mountain  be  known,  and  the  bulk  of  the  earth,  this  experi- 
ment affords  a  ready  mode  of  determining  the  mass,  and  conse- 
quently the  density  of  the  earth  ;  for  the  plumb-line  will  point 
out  the  direction  of  the  resultant  of  two  forces,  one  of  which  is 
the  attractive  force  of  the  earth,  the  other  that  of  the  mountain  ; 
and  the  angle  of  deviation  will  enable  us  to  calculate  one  of  these, 
when  the  other  is  given,  upon  the  principles  of  the  composition 
and  resolution  of  forces  in  §12. 

Professor  Playfair  performed  the  geological  examination  that 
was  necessary  to  determine  the  nature  of  the  materials  com- 
posing the  mountain.  Professor  Button,  upon  these  data,  pro- 
ceeded to  calculate  the  mean  density  of  the  earth,  which  he 
found,  by  his  first  calculation,  to  be  4. 56,  the  density  of  water 
being  taken  as  the  unit ;  but  which,,  on  a  careful  revision  of  the 
process,  he  has  increased  to  5. 

Now  as  the  surface  of  a  large  part  of  the  globe  is  covered  with 
water,  and  the  density  of  the  earthy  matter,  that  forms  by  far  the 
gratest  portion  of  the  residue,  is  little  more  than  twice  as  great  as 
water,  it  becomes  evident,  that  the  earth  is  denser  within  than  at 
the  surface  ;  and  as  we  can  detect  no  sudden  increase  of  density, 
it  is  probable  that  the  variation  in  this  respect  is  regular. 

Observations  on  the  attraction  of  mountains,  made  in  the  neigh- 
bourhood of  Marseilles,  by  the  Baron  de  Zach,  give  analogous 
results  ;  and  we  shall  hereafter  cite  experiments,  made  with  the 
pendulum,  that  corroborate  their  accuracy. 

90.  The  mutual  attraction  of  bodies  nenr  the  surface  of  the 
earth,  has  been  detected  by  Cavendish,  and  his  observations  have 
furnished  another  determination  of  the  mean  density  of  the  earth. 
The  apparatus  employed  by  him  is  admirably  suited  to  the  pur- 
pose for  which  it  was  intended.  It  was  originally  planned  by 


Book  ///.]  OF    SOLID    BODIES.  87 

Mitchell,  a  member  of  the  Rbyal  Society  of  London.  This  ex- 
perimenter being  prevented  by  illness,  which  proved  fatal,  from 
completing  his  researches,  left  by  will  his  apparatus  to  Francis 
I.  H.  Wollaston,  and  from  him  it  passed  into  the  hands  of  Caven- 
dish. The  principle  of  this  apparatus  may  be  explained  as  fol- 
lows : 

If  a  bar  of  an  inflexible  substance  be  accurately  poised  by  its 
middle,  in  a  horizontal  position,  by  means  of  a  thread  or  wire,  the 
nature  of  the  thread  or  wire  is  such  as  to  bring  it  to  rest  in  one 
particular  position.  A  small  force  will  be  sufficient  to  withdraw 
the  bar  from  this  position,  but  the  twisting  or  torsion  which  this 
deflection  will  cause  in  the  wire,  will  gradually  oppose  an  in- 
creasing resistance,  until  this  latter  exceed  the  deflecting  force ; 
the  torsion  will  then  cause  the  bar  to  return  to  its  original  posi- 
tion, whence  the  deflecting  force  will  again  compel  it  to  move. 
Hence  the  bar  will  oscillate  between  two  points,  determined  by 
the  intensity  of  the  deflecting  force,  and  that  of  the  torsion  of 
the  wire.  The  rapidity  of  the  oscillations  will,  upon  principles 
that  we  shall  hereafter  explain,  furnish  a  measure  of  the  intensity 
of  the  deflecting  force. 

Now  if  bodies  mutually  attract  each  other,  a  considerable  mass 
of  a  heavy  substance,  of  a  metal  for  instance,  placed  in  the  same 
horizontal  plane  with  the  bar,  ought  to  cause  a  deflection,  and 
consequent  oscillation  ;  for,  in  this  position,  the  attractive  force  of 
the  Earth  is  completely  counteracted  by  the  exact  poising  of  the 
bar ;  hence  that  influence,  which  would  in  most  other  cases  cloak, 
and  render  imperceptible,  a  force  so  much  smaller  as  that  exerted 
by  the  heaviest  masses  we  have  it  in  our  power  to  move,  would 
become  neutralized. 

91.  Such  being  the  principle,  we  shall  proceed  to  describe  the 
apparatus  more  in  detail.  It  is  represented  in  the  figures  an- 
nexed ;  S  and  S'  are  the  two  spheres  of  metal,  weighing  each  350  Ibs ; 


s- 


88  0*    THE    EQUILIBRIUM  [Book  III. 

a  box  is  represented  in  which  the  bar  is  shut  up,  in  order  to  pre- 
serve it  from  the  action  of  currents  of  air  ;  s  and  s'  are  two  small 
balls  suspended  from  the  extremities  of  the  moveable  bar,  and  by 
whose  weight  it  is  kept  exactly  balanced  and  in  equilibrio. 

The  folio  wing  figure  is  a  horizontal  section  of  the  same  apparatus. 


In  the  previous  figure,  will  be  seen  the  manner  in  which  the  two 
small  balls  are  suspended  from  the  bar  by  a  silver  wire  ;  this  wire 
passes  through  the  bar,  and  is  attached  to  the  vertical  wire,  f\  the 
tenacity  of  the  latter  is  just  sufficient  to  bear,  without  risk  of  break- 
ing, the  weight  of  the  bar  and  the  two  balls  ss'9  and  its  torsion  is 
the  only  force  that  opposes  their  oscillation.  The  two  masses  S 
and  S',  are  themselves  supported  by  iron  rods,  attached  to  a  hori- 
zontal arm  in  such  a  manner  as  to  have  each  a  free  motion  in  a  se- 
micircular arc,  one  on  each  side  of  the  box :  they  can  be  placed 
in  either  of  the  positions  represented  in  the  second  figure,  by  an 
operation  that  is  performed  on  the  outside  of  the  chamber  in 
which  the  apparatus  is  enclosed.  This  chamber  had  neither  doors 
nor  windows,  and  was  illuminated  by  means  of  a  lamp,  which 
was  placed  without  the  chamber,  in  order  that  the  interior  might 
not  be  heated.  An  aperture  opposite  to  the  place  of  the  lamp, 
admitted  the  light  to  fall  upon  one  end  of  the  bar,  and  the  oscil- 
lations were  noted  by  means  of  a  small  telescope,  passed  through 
the  wall,  just  beneath  this  opening. 

The  whole  apparatus  being  at  rest,  and  the  masses  SS'  in  a  po- 
sition at  right  angles  to  the  bar,  they  were  turned  around  until 
they  reached  the  position  represented  in  the  second  figure.  As  soon 
as  this  was  done,  the  bar  began  to  move,  and  the  balls  ss'  to  oscil- 
late. 

The  amount  of  these  oscillations  furnishes  a  measure  of  the 
attraction  of  the  masses  SS',  at  a  given  distance,  and  this  being 
known,  and  compared  with  the  effects  of  the  attraction  of  gravi- 
tation, exerted  by  the  earth,  a  simple  statement  in  proportion 
will  give  the  mass  of  the  latter.  Its  density  was  hence  calculated 
by  Cavendish  to  be  5.48. 

Hutton  has  detected  an  inacuracy  in  the  calculations  of  Cavendish, 
by  correting  which  the  result  has  been  lessened,  and  brought  more 
nsar  to  the  inference  drawn  from  the  experiments  of  Schehallien. 


Book  III.]  OP    SOLID    BODIES.  89 

Still,  the  difference  is  greater,  between  the  results  of  the  two 
methods,  than  it  ought  to  be.  The  same  distinguished  mathema- 
tician has.  therefore,  proposed  that  an  experiment  similar  to  that 
of  Schehallien,  be  repeated  on  the  opposite  faces  of  the  great 
Egyptian  pyramid,  at  the  height  of  one  fourth  from  the  base.  The 
regularity  of  the  figure  of  this  great  artificial  mass,  and  its  proba- 
ble identity  of  composition  throughout,  render  this  method  so 
accurate,  as  in  all  probability  to  settle  this  disputed  question. 
Until  this  be  done,  we  may  without  much  risk  of  error,  assume  that 
the  mean  density  of  the  earth  is  five  times  as  great  as  that  of  water. 

92.  Knowing  the  density  of  the  earth  when  compared  with 
water  as  the  unit,  the  weight  of  a  given  bulk  of  water,  and  the 
volume  of  the  earth,  we  may  proceed  to  calculate  its  weight,  in 
the  units  of  a  conventional  system  of  measures.     Astronomic 
observations,  and  the  calculations  founded  upon  them,  enable  us 
to  compare  the  mass  of  the  earth  with  that  of  the  sun  and  moon; 
knowing  thus  the  mass  of  the  sun,  we  calculate  readily  the  mass 
of  any  planet  accompanied  by  a  satellite;  and  the  determination 
of  the  mass  of  those  planets  that  have  no  satellites,  although  less 
easy,  is  also  practicable  ;  thus,  as  has  been  well  remarked,  the 
apparatus  of  Cavendish  is  in  fact  a  balance,   in  which  we  can 
weigh  the  vast  bodies  that  compose  the  solar  system. 

93.  As  bodies  fall  to  the  earth  when  left  without  support,  at 
all  places  within  our  reach,  it  may  be  inferred  that  the  attraction 
of  gravitation  acts  during  the  whole  time  of  their  descent,  and  is 
therefore  an  accelerating  force.    This  is  also  conclusively  shown, 
by  the  well  known  fact  of  the  acceleration  that  takes  place  in  the 
motion  of  falling  bodies  ;  these  strike  the  earth  with  greater  velo- 
city, in  proportion  as  the  distance  through  which  they  have  fallen 
increases.     We  cannot  however,  have  recourse  to  experiments 
of  this  kind,  in  order  to  discover  whether  this  accelerating  force 
be  constant,  or  variable  in  its  intensity  ;  for,  as  the  air  causes  an 
unequal  velocity  in  bodies  of  unequal  densities,  and  prevents  the 
fall  of  some   altogether,  and  as  we  cannot  obtain  a  vacuum  of 
sufficient  extent  for  the  experiment,  we  cannot  receive  the  indi- 
cations obtained  by  the  fall,  even  of  the  densest  bodies,  as  abso- 
lutely accurate. 

The  resistance  of  the  air,  as  will  be  hereafter  seen,  varies  with 
the  square  of  the  velocity  ;  hence,  if  the  velocity  of  a  falling  body 
be  diminished  in  an  arithmetic  progression,  the  resistance  of  the 
air  decreases  in  geometric  ratio.  Could  we  therefore  observe 
the  motion  of  a  body  actuated  by  gravity,  under  circumstances 
where  its  motion  would  be  diminished,  without  any  alteration  ta- 
king place  in  the  law  of  the  accelerating  force,  we  might  obtain 

12 


90  OF  THE    EQUILIBRIUM  [Book  III 

that  law,  almost  without  its  being  affected  by  the  resistance  of 
the  air. 

94.  We  may  consider  the  force  of  gravity  as  acting  parallel  to 
itself  within  small  horizontal  distances;  a  body  therefore,  moving 
upon  a  plane  slightly  inclined  to  the  horizon,  will  suit  our  pur- 
pose ;  for,  by  §56,  the  force  by  which  a  body  is  actuated,  when 
it  moves  on  a  plane,  inclined  to  the  direction  of  an  accelerating 
force  at  a  constant  angle,  follows  the  same  law  with  the  accelera- 
ting force  itself.  That  is  to  say,  that  although  the  absolute  velo- 
city is  lessened,  the  relation  between  the  spaces  described,  during 
different  portions  of  the  time  of  the  motion's  continuance,  is  con- 
stant. Hence,  should  we  find  that  the  velocity  with  which  a 
body  descends  on  an  inclined  plane  is  uniformly  accelerated,  we 
may  infer,  that  the  force  of  gravity  is  constant,  within  the  limits 
of  the  plane's  elevation. 

Experiments  were  made  by  Galileo  upon  this  principle.  He 
formed  his  inclined  plane  by  stretching  a  cord  twenty  or  thirty 
feet  in  length,  between  the  two  fixed  points,  at  different  levels.  The 
difference  of  height  between  the  two  points  must  be  small,  and 
the  inclination  of  the  plane  is  therefore  small  also  ;  the  velo- 
cities were  diminished,  according  to  the  principle  in  §56,  in  the 
ratio  of  the  sine  of  the  plane's  inclination  to  the  horizon.  In  or- 
der to  lessen  the  friction,  the  bodies  were  mounted  upon  wheels, 
and  the  smallness  of  the  velocities  rendered  the  resistance  of  the 
air  insensible.  Experiments  made  with  this  apparatus,  showed 
an  uniform  acceleration  in  the  velocity  of  the  bodies  ;  and  hence, 
that  gravity  was,  within  the  limits  occupied  by  the  plane,  a  con- 
stant accelerating  force. 

95.  The  machine  of  Atwood  affords  a  more  convenient  and 
elegant  method  of  obtaining  the  same  results. 

Two  bodies,  of  unequal  weights,  are  united  by  a  cord  passing 
over  a  pulley.  It  is  therefore  obvious  that  the  preponderance 
of  the  one  will  cause  it  to  descend,  and  raise  the  other,  through 
an  equal  space  ;  but  it  is  also  obvious,  that  this  motion  cannot  be 
as  rapid,  as  that  which  either  body  would  have,  if  free.  The  re- 
lation which  the  accelerating  force,  in  such  a  system,  bears  to 
the  whole  force  of  gravity  may  be  easily  investigated. 

Let  A  and  B  be  the  two  bodies,  and  g  the  force  of  gravity, 
which  is  the  accelerating  force  that  acts  upon  them  both,  but 
whose  effects  are  modified  by  their  mutual  action. 

Let  t  be  the  time  elapsed  since  the  motion  began ;  v  the  velo- 
city of  A  ;  v'  the  velocity  of  B. 

These  velocities  are  obviously  equal,  but  in  contrary  directions. 

During  the  time  dt,  the  velocities  will  increase  under  the  action 
of  the  accelerating  force,  by  quantities  which  will  be  represented 
by  dv  and  dv' ;  during  the  same  time  the  bodies,  if  falling  freely, 


Book  III.]  or  SOLID  BODIES.  »1 

would  each  acquire  (§49)  the  velocity  gdt.  Hence  the  body  A 
will  lose,  in  consequence  of  its  connexion  with  B,  a  velocity  rep- 
resented by  gdt  —  dv  ;  and  B  will  lose  by  its  connexion  with  A,  a 
velocity  represented  by  gdt  —  dv.  These  are  respectively 
equal  to  a"  and  6"  in  the  formula  (98)  ,  Aa"+B&"+&c.=0,  that 
expresses  the  principle  of  D'Alembert,  while  A  and  B  are  res- 
pectively equal  to  the  quantities  of  the  same  name  ;  and  the  mo- 
tion of  the  bodies  being  contrary,  the  second  term  becomes  nega- 
tive ;  therefore,  by  substitution,  we  have 

A(gdt—  dv}—  Bfecfc—  cfo')=0  ; 
and  as  the  velocities  are  equal,  but  with  contrary  signs, 

dv=dv  ; 
therefore, 

A(gdt—  dt>)—  B(gefc+dv)=0  ; 
performing  the  multiplications,  we  obtain 

Agdt—Bgdt—  Adv—  Bdv=0  ; 
by  transposition', 

(A— 
dividing  by  A-J-B, 


integrating, 

A—  B 


in  this  expression  the  arbitrary  constant,  a,  represents  the  initial  ve- 
locity, which  when  the  bodies  move  from  rest,  becomes  equal  to 
0  ;  in  which  case 


The  forces  which  impel  equal  moving  bodies,  or  systems  of 
bodies,  being  proportioned  to  the  velocities,  we  obtain  the  fol- 
lowing law. 

The  force  which  remains  to  cause  the  descent  of  the  heavier 
body,  in  Atwood's  machine,  is  to  the  whole  force  of  gravity,  as 
the  difference  of  the  weights  of  the  two  bodies  is  to  their  sum. 

Experiments  with  this  apparatus,  show  a  motion  uniformly 
accelerated  ;  hence  we  have  a  farther  proof  that  the  attraction  of 
gravitation  is  a  constant  force,  within  the  space  occupied  by  the 
machine. 

As  the  formula  we  have  investigated,  gives  the  relation  between 
the  velocity  the  body  A  has,  when  united  to  B,  and  that  which 
it  would  havs  when  falling  freely,  this  machine  gives  us  a  ready 
mode  of  obtaining  the  velocity  of  falling  bodies.  Thus  if  the 
body  A  have  a  weight  of  101,  and  B  of  99,  their  sum  will  be 
200,  their  difference  2,  and  the  proportion  of  the  actual  velocity, 
to  that  obtained  by  falling  freely  Tfo.  Now  a  velocity  no  more 


93  O*    THE   EQUILIBRIUM  [Book  HI. 

than  Tioth  part  of  that  acquired  by  a  falling  body,  is  so  slow,  that 
it  will  be  perfectly  easy  to  note  the  marks  upon  a  vertical  scale 
attached  to  the  instrument  with  which  the  descending  weight 
corresponds,  at  each  beat  of  the  clock  that  is  also  attached.  By 
such  experiments,  carefully  conducted,  it  is  found  that  the  actual 
descent,  in  the  instance  we  have  stated,  is  1.93  inches  in  the  first 
second  ;  whence  the  fall  of  a  heavy  body  from  rest  may  be  infer- 
red to  be  193  in.  in  the  first  second  of  time,  or  16  feet  and  an  inch. 

The  machine  of  Atwood  is  represented  on  the  opposite  page. 

In  this  will  be  seen, 

1.  The  pulley,  «,  the  extremities  of  whose  axles  each  rest  upon 
two  other  wheels,  bb,  the  axles  of  which  rest  on  agate  planes;  in 
this  manner,  as  will  hereafter  be  shown,  the  friction  is  reduced 
to  the  smallest  possible. 

2.  A  divided  scale  along  which  the  body  A  descends  without 
touching  it.  On  thisscale  are  placed  two  moveable  plates/and  g, 
one  pierced  in  the  form  of  a  ring,  the  other  a  plane  surface.     The 
weights  are  at  first  equal,  and  the  preponderance  of  A  is  effected, 
by  laying  upon  it  a  body  of  the  shape  of  a  bar,  which  cannot  pass 
through  the  ring;  thus  the  accelerating  force  may  be  removed 
at  any  point,  and  the  velocity  will  become  uniform  ;  whence  not 
only  the  spaces  described  by  the  accelerated  motion,  but  the  final 
velocities  also,  may  be  made  the  subjects  of  experiment. 

3.  In  order  to  count  the  time,  during  which  the  motion  takes 
place,  a  clock  beating  seconds  is  attached  to  the  side  of  the  ma- 
chine, and  a  spring  is  adapted,  in  such  a  way,  that  the  weight  A, 
and  the  pendulum  P,  of  the  clock  E,  may  be  released  at  the  same 
instant  of  time. 

Applied  to  ascertain  the  final  velocities,  this  machine  gives  the 
results  obtained  from  theory  in  §  56,  namely,  that  the  velocities 
acquired  by  moving  from  rest,  under  the  action  of  an  accelerating 
force,  are  such  as  to  carry  the  body  with  uniform  motion  through 
twice  the  space,  in  a  time  equal  to  that  employed  in  acquiring 
those  velocities. 

96.  The  inferences  obtained  from  the  experiments  made  with 
the  inclined  plane  of  Galileo,  and  the  machine  of  Atwood,  have 
been  purposely  restricted  to  the  limits  occupied  by  the  respective 
apparatus ;  for  it  will,  even  at  first  sight,  appear  probable,  that 
the  force  of  gravity,  residing  in  the  mass  of  the  earth,  although 
apparently  constant  within  small  limits,  must  decrease  as  we 
recede  from  the  earth,  according  to  some  law  dependent  upon,  or 
in  mathematical  terms,  according  to  some  function  of  the  dis- 
tance. 

Galileo,  who  first  ascribed  the  fall  of  heavy  bodies  to  a  me- 
chanical force  exerted  upon  them  by  the  earth,  did  not  attempt  to 


Book  ///.] 


OP    SOLID    BODIES. 


94  OP    THE    EQUILIBRIUM  [  &00k  III. 

extend  the  action  of  that  force  to  bodies  not  in  the  immediate 
vicinity  of  the  earth.  Kepler,  with  more  extended  views,  saw 
that  it  could  not  be  thus  limited  in  its  action,  but  failed  in  disco- 
vering the  law,  according  to  which  its  intensity  varies,  in  terms 
of  the  distance.  His  views  were  in  consequence  neglected,  and 
almost  completely  forgotten. 

Newton,  like  Kepler,  saw  that  the  action  of  gravity,  which  does 
not  diminish  in  a  degree  that  it  had  been  possible  previous  to  his 
time  to  detect,  even  at  the  top  of  the  highest  mountains,  could  not 
cease  suddenly.  He  hence  inferred  that  it  might  extend  to  the 
moon,  and  be  the  cause  by  which  that  body  was  compelled  to 
describe  a  re-entering  orbit.  But  although  it  might  extend  to 
the  moon,  it  was  improbable  that  it  could  act  there,  with  an  inten- 
sity equal  to  that  it  exerts,  at  the  surface  of  the  earth.  The  direc- 
tions of  gravity,  supposing  the  earth  to  be  spherical,  are  directed 
to  the  centre  :  hence  the  influence,  whatever  be  its  cause,  is  a  radi- 
ating one,  and  would  probably  follow  the  same  law  in  its  decrease, 
as  the  radiating  actions  of  light  and  heat.  The  force  of  these 
natural  agents,  is  well  known  to  vary,  in  the  inverse  ratio  of  the 
squares  of  the  distances  from  the  centre.  Newton  inferred,  that 
the  decrease  in  the  intensity  of  gravity,  must  follow  the  same 
law;  and  that  at  the  moon,  the  attractive  force  must  be  as  much 
less  than  it  is  at  the  surface  of  the  earth,  as  the  sqliare  of  the 
radius  of  the  earth  is  less  than  the  square  of  the  moon's  distance 
from  the  earth's  centre. 

The  moon,  if  not  acted  upon  by  a  force  drawing  it  to  the  earth, 
would,  according  to  the  principles  of  motion,  §  62,  go  off  in  the  di- 
rection of  a  tangent.  Taking  the  motion  of  the  moon  for  a  short 
space  of  time,  and  consequently  in  a  small  arc,  the  tangent  of  the 
arc  will  represent  the  force  with  which  the  moon  would  proceed 
in  a  rectilineal  direction,  the  space  between  the  tangent  and  the 
extremity  of  the  arc,  will  represent  the  measure  of  the  deflecting 
force.  If  the  arc  be  taken  which  corresponds  to  the  mean  motion 
of  the  moon  for  a  minute  of  time,  this  space  may  be  calculated  to 
be  about  sixteen  feet.  Now  the  measure  of  the  force  of  gravity, 
g-,  at  the  surface  of  the  earth  is  twice  the  space  a  body  falls  from 
a  state  of  rest  in  a  second  of  time,  or  is  in  the  nearest  round  num- 
bers 32  feet.  The  mean  distance  of  the  moon  is  about  60  semi- 
diameters  of  the  earth,  and  the  respective  distances  of  the  two 
bodies  have  the  ratio  of  60  to  1.  assuming  with  Newton,  that  the 
force  varies,  in  the  inverse  ratio  of  the  squares  of  the  distances, 
the  value  of  gfa  the  force  of  gravity  at  the  moon,  is 

32 


the  space  described  in  a  given  time,  under  the  action  of  a  constant 
accelerating  force,  and  within  this  small  space  the  force  may  be 
considered  constant,  is  by  the  formula  (61) 


///.]  OP    SOLID    BODIES.  95 


taking  the  value  of  g'  given  above  as  the  measure  of  g  the  accele- 
rating force  in  this  formula,  and  making 

*=6G", 
or  a  minute  of  time,  we  have 

s=32+W° 
2-f-(60)'J 

The  deflection  of  the  moon  from  a  tangent  to  her  orbit  in  a 
minute  of  time,  is  therefore  the  same  with  that  which  it  should  be, 
were  she  acted  upon  by  an  attracting  force  residing  in  the  centre 
of  the  earth,  and  varying  in  its  intensity,  in  the  inverse  ratio  of  the 
squares  of  the  distances. 

By  an  investigation  analogous  to  this,  Newton  inferred  that 
the  attractive  force  of  the  earth  did  not  cease  suddenly,  but  ex- 
tended as  far  as  the  orbit  of  the  moon,  and  acted  there  according 
to  the  law  observed  in  other  forces,  or  influences,  emanating 
from  a  centre. 

97.  The  experiments  that  have  been  cited  §  91,  showing  a  mu- 
tual attraction  between  gravitating  bodies,  it  may  readily  be  infer- 
red, that  the  moon  has  an  action  upon  the  earth  proportioned  to  its 
mass.  This  attraction  of  the  moon  may  be  shown  by  its  influ- 
ence upon  the  liquid  mass  of  the  ocean,  in  which  it  forms  waves, 
that  constitute  an  important  part  of  the  tides  :  it  is  also  obvious 
in  various  astronomic  phenomena  that  are  foreign  to  our  subject. 

9S.  It  being  thus  found  that  the  attraction  is  mutual  between 
the  earth  and  moon,  analogy  leads  us  to  infer  that  the  earth  and 
sun  mutually  attract  each  other,  with  forces  proportioned  to  their 
respective  masses,  and  that  the  same  mutual  action  takes  place 
between  all  the  bodies  that  compose  the  solar  system.  The  as- 
tronomy of  observation  fully  confirms  this  important  truth  ;  anoT 
physical  astronomy,  thus  established  on  a  secure  basis,  applies 
the  principles  of  mechanics  to  investigate  the  minute  changes 
and  variations  that  this  multitude  offerees,  exerted  by  moving 
bodies,  are  constantly  producing  in  each  other's  motions.  Thus 
the  simple  mechanical  laws,  which  we  are  compelled  to  investigate 
in  order  to  explain  the  action  of  the  most  familiar  machines,,  are 
the  same  that  direct  the  vast  mechanism  of  the  heavens.  Nor  does 
the  research  that  these  laws  give  rise  to,  stop  at  the  bounds  of 
our  own  system.  The  fixed  stars  also  obey  the  same  universal 
force;  and  the  astronomers  of  Europe  are  now  on  the  threshold 
of  discoveries  in  respect  to  them,  more  wonderful  than  even  those 
with  which  Laplace  has  closed  the  labour  began  by  Newton,  leav- 
ing no  motion  in  our  planetary  system,  that  is  not  reducible  to 
mechanical  principles. 


06  OF   THE    EQUILIBRIUM  [Book  III. 

99.  The  law  according  to  which  gravity  decreases,  shows  us, 
that  although  to  all  appearance  constant  within  the  limits  of  our 
experiments,  it  is  not  absolutely  so  ;  and  we  have  at  present 
modes  of  observation  that  will  hereafter  be  noticed,  by  which 
even  the  small  decrease  that  occurs  in  distances  within  our  reach, 
may  be  rendered  evident. 

100.  Were  the  earth  at  rest  in  space,  and  perfectly  spherical 
in  its  form,  the  force  of  gravity  would  be  constant  at  every  point 
of  its  surface  ;   but  if  it  be  in  motion  this  cannot  be  the  case. 

When  a  body  revolves  around  a  fixed  axis,  its  several  points 
describe  circles  whose  planes  are  perpendicular  to,  and  their  cen- 
tres in,  the  axis.  From  §  t>4,  it  would  appear  that  these  points, 
revolving  in  equal  times,  are  influenced  by  centrifugal  forces 
that  are  proportioned  to  the  radius  of  the  circles  they  respect- 
ively describe  ;  and  the  directions  of  the  forces,  are  in  the  planes 
of  these  circles.  Now  these  forces  must  lessen  the  action  of  the 
force  of  gravity,  which  is  due  to  the  attraction  of  the  whole 
mass  of  the  earth,  but  they  will  affect  it  differently  in  different 
latitudes.  At  the  poles,  the  centrifugal  force  is  equal  to  0,  for 
the  circle  becomes  a  mere  point;  while  it  is  greatest  in  points 
situated  in  the  equator,  as  that  is  the  greatest  circle  of  diurnal 
rotation.  At  the  equator,  too,  the  centrifugal  force  acts  in  direct 
opposition  to  the  force  of  gravity,  while  in  all  other  places  they 
are  inclined  to  each  other. 

The  law  which  the  decrease  of  gravity,  influenced  by  these 
two  circumstances,  follows,  upon  the  surface  of  the  earth,  between 
the  poles  and  the  equator,  may  be  thus  investigated  : 

The  general  expression  for  the  centrifugal  force  is  (83) 


call  the  diminution  of  gravity  at  any  latitude/7  ; 
the  radius  of  the  circle,  described  by  any  point  on  the  surface,  is 
equal  to  the  cosine  of  the  latitude  ;  the  time  T  is  a  sidereal  day, 
and  is  constant  ;  substituting  the  former  value,  and  calling  the 
latitude  L,  we  have 

_4^cos.  L 
/=     —  Tl         . 

but  this  force  does  not  act  in  direct  opposition  to  the  force  of 
gravity,  the  one  being  parallel  to  the  equator,  while  the  other  is 
directed  to  the  centre.  We  must  therefore  resolve  the  centrifugal 
force  f  into  two  others,  one  of  which  is  parallel,  and  the  other 
perpendicular  to  the  surface  of  the  earth,  at  the  given  latitude  ; 
the  latter  is  that  which  acts  to  diminish  the  force  of  gravity,  and 
may  be  found  by  the  formula  of  the  Resolution  offerees  (11) 
X=R  cos.  a, 


Book  fflj  OP  SOLID  BODIES.  97 

hence  the  expression  for  the  value  of  the  centrifugal  force  must  be 
multiplied  by  the  cosine  of  the  angle  its  direction  makes  with  the 
plumb-line,  or  by  the  cosine  of  the  latitude,  and  the  value  of  the 
diminution  of  gravity,  at  any  latitude,  becomes 


If  then  the  earth  were  a  perfect  sphere,  the  centrifugal  force 
at  the  equator  would  bear  to  the  force  that  lessens  the  attraction 
of  gravity  at  any  latitude,  the  ratio  of  the  radius  of  the  earth 
to  the  square  of  the  cosine  of  the  latitude. 

101.  The  ratio  of  the  centrifugal  force  at  the  equator,  to  the 
whole  force  of  gravity,  may  be  ascertained  in  the  following 
manner  : 

Let  G  be  the  whole  attractive  force  of  the  Earth  at  the  sur- 
face, g  the  apparent  force  of  gravity  at  the  Equator,  then 

(106) 

In  this  expression,  G  is  twice  the  space  a  body  descends  from 
rest  in  a  second  of  time,  or 

G=32.16feet. 

and,  flr=3.1416; 

r  is  the  Earth's  radius,  which  calculated  in  feet,  gives 

r  =41,  836420  feet; 
the  sidereal  day  reduced  to  seconds  gives 

T=86164"; 
substituting  these  values, 

#=32.16—0.1118;  (107) 

and  as 

32.16 


0.1118"288' 


therefore — 


The  ratio  of  the  centrifugal  force  at  the  equator  is  to  the  ap- 
parent force  of  gravity  as  1  to  288  ;  or  the  diminution  of  gra- 
vity, between  the  poles  and  the  equator,  is  aigth  part  of  the 
whole. 

102.  These  relations  have  been  investigated  upon  the  hypo- 
thesis of  the  earth's  being  a  sphere,  but  this  could  only  occur  in 
a  moving  body,  in  the  case  of  its  being  a  solid  mass  devoid  of 
all  elasticity.  Every  point  in  a  moving  spherical  body,  being 
influenced  by  a  centrifugal  force,  greatest  at  the  Equator  and  least 
at  the  Poles,  these  points  would  tend  to  assume  a  state  of  equi- 

13 


08  O*  THE    EQUILIBRIUM  [JBook  llL 

librium,  under  the  joint  action  of  the  centrifugal  and  gravitating 
forces.  If  then  the  mass  were  originally  spherical,  the  equato- 
rial parts  would  tend  to  recede  from,  the  polar  to  approach  the 
centre ;  and  were  they  free  to  move,  the  equatorial  diameter  would 
be  increased,  and  the  polar  would  be  diminished,  until  a  state 
of  equilibrium  were  obtained.  Now  although  we  know  nothing 
certain,  in  respect  to  the  state  in  which  the  interior  of  the  earth 
exists,  and  find  its  outer  crust  a  solid  body,  yet  in  large  basins  or 
cavities  of  that  crust,  a  fluid,  (the  ocean,)  exists,  covering  nearly 
three-fourths  of  the  surface.  The  level  of  this  mass  of  fluid  points 
out,  as  has  been  seen,  the  mean  surface  of  the  earth.  Now  this 
fluid  mass  could  only  be  in  a  permanent  state,  if  the  general 
shape  of  the  crust  had  the  form  the  fluid  would  itself  assume. 
Hence  in  whatever  state  the  mass  of  the  Earth  may  have  been 
originally  created,  its  shape  is  that  of  a  fluid  body  retained  in 
equilibrio,  by  the  joint  action  of  the  centrifugal  force  and  that  of 
gravity.  This  form  has  been  investigated  under  a  variety  of  hy- 
potheses, and  by  different  persons.  It  is  not  our  province  to  enter 
into  these  investigations;  we  shall  therefore  content  ourselves 
with  simply  stating  a  few  of  the  results. 

Newton,  considering  the  earth  as  a  homogeneous  fluid  mass, 
endued  with  a  rotary  motion,  and  composed  of  particles,  attract- 
ing each  other,  according  to  his  law  of  universal  gravitation,  took 
it  for  granted,  that  it  would  assume  the  figure  of  an  oblate  sphe- 
roid. With  these  data,  he  inferred  that  the  flattening  of  such  a 
spheroid  would  be  fths  of  the  relation  of  the  centrifugal  force  at 
the  equator,  to  the  whole  force  of  gravity.  The  latter  has  been 
shown  to  be  ^9  and  hence  the  ratio  between  the  equatorial  and 
polar  axes  would  be,  upon  his  hypothesis,  230  :  229. 

Huygens,  assuming  the  whole  force  to  reside  in  the  centre, 
determined  the  ratio  to  be  578  :  577. 

The  last  result  has  been  shown  since,  to  be  consistent  with  the 
theory  of  mutual  attraction,  under  the  hypothesis,  that  the  earth  is 
infinitely  dense  at  the  centre,  and  infinitely  rare  at  the  surface. 
Now  as  we  have  shown,  (§  91)  that  the  mean  density  of  the  earth 
is  greater  than  that  of  its  surface,  and  hence  inferred,  that  the  den- 
sity increases  towards  the  centre,  the  figure  of  the  earth  must 
vary  between  these  limits,  and  the  oblateness  of  the  terrestrial 
spheroid  cannot  be  greater  than  ^^  nor  less  than  3^¥. 

Such  being  the  theory,  it  will  be  obvious,  that  if  a  flattening  at 
the  poles  can  in  any  manner  be  detected,  such  flattening  would 
furnish  conclusive  evidence  of  the  diurnal  motion  of  the  earth. 
The  same  would  be  shown  by  a  decrease  in  the  apparent  action 
of  gravity,  between  the  poles  and  the  equator. 


Book  III.}  OF    SOLID    BODIES.  09 

Both  of  these  may  be  ascertained  to  exist,  by  methods  that  re- 
main to  be  described,  but  the  oblateness  may  and  has  been  de- 
tected by  actual  measurement. 

103.  To  recapitulate  the  laws  of  Universal  Gravitation. 
(1.)  It  is  common  to  all  bodies,  and  mutual  between  them. 
(2.)  It  is  proportioned  to  the  quantity  of  matter  in  the  body. 
(3.)  Its  intensity  decreases,  as  the  square  of  the  distance  from 
the  centre  of  attraction  increases. 


100  OF    THE    CENTRE  [Book  111. 

CHAPTER  III. 
OF  THE  CENTRES  OF  GRAVITY  AND  INERTIA. 

104.  From  what  has  been  said  in  the  last  chapter,  on  the  sub- 
ject of  the  Attraction  of  Gravitation,  it  appears,  that  bodies  near 
the  surface  of  the  earth,  being,  like  all  others,  influenced  by  it, 
may  be  considered  as  acted  upon  by  a  number  of  gravitating 
forces,  tending  to  draw  each  particle,  or  material  point,  in  their 
mass,  towards  the  centre.     These  forces,  within  the  space  occu- 
pied by  even  the  largest  bodies,   may  be  considered  as  equal  ; 
and  although  their  directions  actually  converge,  yet  in  any  given 
body  the  convergence  is  insensible  ;  no  error  can  therefore  arise 
from  considering  them  as  absolutely  parallel. 

A  heavy  body,  near  the  surface  of  the  earth,  may  therefore 
be  considered  as  acted  upon  by  a  number  of  equal  and  parallel 
forces.  These  forces  have  a  resultant,  which  is  equal  to  their 
sum,  and  is  identical  with  the  weight  of  the  body.  This  weight 
will  depend  upon  the  volume  of  the  body,  its  density,  and  the 
intensity  of  gravity  at  the  place  in  which  it  is  situated.  If  we 
call  the  volume  or  bulk  B,  the  density  D,  and  the  measure  of 
the  force  of  gravity  g,  the  value  of  the  weight  W  will  be 

W=B  D  g, 

and  as  by  (101)  the  mass,  M=B  D, 

W=M^. 

105.  The  point  of  application  of  this  resultant,  which,  in  the 
abstract  theory  of  parallel  forces,  has  been  called  their  Centre,  is 
in  the  case  of  gravitating  bodies  called  the  Centre  of  Gravity. 

The  formulae  then  of  §  16,  by  means  of  which  the  centre  of 
parallel  forces  is  found,  and  the  several  inferences  obtained  from 
those  formulae  in  particular  cases,  are  applicable  to  the  subject 
before  us.  So  also  are  the  inferences  from  the  geometric  inves- 
tigations, in  the  same  section.  That  this  is  true  in  respect  to  ho- 
mogeneous solid  bodies  is  evident,  for  they  may  be  considered 
as  made  up  of  a  number  of  equal  particles,  uniformly  distributed 
throughout  the  mass.  In  practice,  however,  it  frequently  be- 
comes necessary  to  determine  the  centres  of  gravity  of  lines  and 
surfaces,  and  even  of  abstract  figures  of  three  dimensions.  For 
this  purpose,  they  are  supposed  to  be  divided  into  an  infinite  num- 
ber of  small  and  equal  parts,  each  of  which  is  influenced  by  an 
equal  gravitating  force. 


Book  ///.]  OP    GRAVITY.  1 01 

The  inferences  before  obtained,  may  be  now  recapitulated,  in 
reference  to  this  individual  case  of  parallel  forces. 

(1.)  The  centre  of  gravity  of  a  straight  line  bisects  it. 

(2.)  The  centre  of  gravity  of  two  straight  lines  is  found  by 
joining  the  two  points  that  bisect  them,  and  dividing  the  line 
that  joins  them,  into  parts  reciprocally  proportional  to  the  mag- 
nitudes of  the  two  lines.  Of  three  lines,  the  centre  of  gravity 
may  be  found,  by  first  finding  the  centre  of  gravity  of  two  of 
them  ;  this  is  then  to  be  joined  by  a  straight  line  to  the  point 
that  bisects  the  third,  and  this  last  line  divided  into  parts  recipro- 
cally proportioned  to  the  joint  magnitude  of  the  two  first,  and 
the  third  line.  The  centre  of  gravity  of  four  lines  may  be  found, 
by  first  finding  the  centre  of  three  of  them,  and  uniting  it  to  the 
point  that  bisects  the  fourth,  which  is  then  to  be  divided  as  in  the 
former  case.  In  this  manner  the  centre  of  gravity  of  the  peri- 
meter of  a  triangle,  or  other  figure,  bounded  by  straight  lines, 
may  be  determined. 

(3.)  The  centre  of  gravity  of  the  surface  of  a  triangle,  is  in 
the  line  that  joins  the  vertex  to  the  point  that  bisects  the  base, 
at  the  distance  of  two-thirds  of  that  line  from  the  vertex. 

(4.)  The  centre  of  gravity  of  a  quadrilateral  figure  may  be 
found,  by  dividing  it  into  two  triangles,  joining  their  respective 
centres  of  gravity,  and  dividing  the  line  that  unites  them  into 
parts  reciprocally  proportional  to  the  area  of  the  two  triangles. 
And  in  general  the  Centre  of  gravity  of  any  polygon  whatever 
may  be  found  by  dividing  it  into  triangles,  the  centre  of  gravity 
of  two  of  which  is  first  found,  and  united  to  the  centre  of  gra- 
vity of  the  third  by  a  line,  that  is  to  be  divided  in  a  similar  ratio ; 
this  point  is  then  to  be  united  by  a  straight  line,  to  the  centre  of 
gravity  of  the  fourth  triangle,  and  so  on.  Any  polygon,  it  is 
well  known,  can  be  divided  into  as  many  triangles,  less  two,  as 
it  has  sides. 

(5.)  The  centre  of  gravity  of  a  circle  is  in  its  centre.  This 
point  is  also  the  centre  of  gravity  of  a  ring  contained  between 
two  concentric  circles. 

The  centre  of  gravity  of  a  circular  arc,  is  at  a  distance  from 
the  centre  of  the  circle,  which  is  a  fourth  proportional  to  the 
lengths  of  the  arc,  the  chord,  and  the  radius  of  the  circle. 

In  a  semicircle  this  distance  is 


=0.63662  r. 


1.5708 

(6.)  The  centre  of  gravity  of  a  parallelogram  is  in  the  point 
where  its  two  diagonals  intersect  each  other. 

(7.)  The  centre  of  gravity  of  a  parabola  is  in  its  axis,  at  the 
distance  of  three-fifths  of  that  line  from  the  vertex. 


103  OP    THE    CENTRE  [Book  HI. 

(8.)  The  centre  of  gravity  of  the  surface  of  a  solid,  formed 
by  the  revolution  of  a  plane  surface  around  a  line,  in  respect  to 
which,  all  its  parts  are  symmetrically  situated,  is  the  same  as 
the  centre  of  gravity  of  the  generating  surface. 

Thus :  the  centre  of  gravity  of  a  hollow  cylinder  is  in  the 
point  that  bisects  its  axis ;  the  centre  of  gravity  of  a  hollow  cone 
is  in  its  axis  at  a  distance  of  two-thirds  the  length  of  that  axis 
from  the  vertex.  The  centres  of  gravity  of  hollow  spheres,  and 
ellipsoids  of  revolution,  are  in  their  centres  of  magnitude. 

(9.)  The  centre  of  gravity  of  the  surface  of  a  spheric  segment 
bisects  its  versed  sine. 

(10.)  The  centre  of  gravity  of  a  triangular  pyramid  is  in  the 
line  that  joins  its  vertex  to  the  centre  of  gravity  of  the  base, 
and  at  the  distance  of  three-fourths  of  that  line  from  the  vertex. 

(11.)  The  centre  of  gravity  of  any  solid  figure,  bounded  by 
plane  surfaces  may  be  found  by  dividing  it  into  a  number  of  tri- 
angular pyramids,  and  proceeding  in  them  as  has  been  directed  to 
be  done  in  regard  to  the  triangles,  into  which  plane  surfaces  are 
divided  for  a  similar  purpose. 

(12.)  The  centre  of  gravity  of  a  solid  cone,  is  in  its  axis  at 
the  Distance  of  three-fourths  of  that  line  from  the  vertex. 

(13.)  The  centre  of  gravity  of  a  sphere  is  in  its  centre  of 
magnitude,  as  is  the  centre  of  gravity  of  a  shell  contained  be- 
tween two  concentric  spheres. 

(14.)  The  centre  of  a  gravity  of  a  solid  paraboloid  is  at  a 
distance  from  the  vertex,  equal  to  two-thirds  of  the  axis. 

(15.)  The  centre  of  gravity  of  a  spheric  segment  is  in  its  fixed 
axis,  at  the  distance  of  fths  of  its  length  from  the  vertex. 

(16.)  The  centre  of  gravity  of  a  cycloid,  that  is  bisected  by 
the  vertex,  is  in  the  diameter  of  the  generating  circle,  at  a  dis- 
tance of  one-third  of  the  perpendicular  height  from  the  vertical  arc. 

106.  In  order  that  a  heavy  body  shall  be  supported,  it  is  ne- 
cessary that  a  force  shall  be  applied  to  it,  equal  in  magnitude,  and 
contrary  in  direction,  to  the  force  of  gravity  that  acts  upon  it. 
The  direction  of  gravity  is  a  vertical  line,  and  its  point  of  appli- 
cation is  the  centre  of  gravity  ;  hence  the  supporting  force  must 
act  perpendicularly  upwards,  and  must  be  applied  either  to  the 
centre  of  gravity  itself,  or  somewhere  in  the  vertical  line  pass- 
ing through  it,  which  is  called  its  Line  of  Direction. 

If  the  supporting  force  be  applied  to  a  single  point  in  the  body, 
there  are  three  cases  that  may  occur  : 

(1.)  In  the  first  place,  the  point  of  support  may  be  above  the 
centre  of  gravity  ;  in  this  case,  the  centre  of  gravity  will  be  in  the 
vertical  line  passing  through,  or  will  be  directly  beneath,  the  point 
of  support ;  this  case  occurs  in  all  instances  of  suspension,  where 


Book  III]  ofr  GRAVil-t.  103 


the  line  of  support  is  vertical^ano!  Trie  centre  of  gravity  is  in  the 
line  of  support  produced  ;  when  this  occurs,  the  centre  of  gra- 
vity is  in  the  lowest  possible  point.  If  the  equilibrium  of  such 
a  system  be  in  any  manner  disturbed,  the  body  will  oscillate  on 
each  side  of  the  vertical  line,  and  if,  as  always  happens  in  nature, 
its  motion  be  opposed  by  resistances,  the  body  speedily  returns 
to  rest  in  its  original  position;  hence  the  equilibrium  is  said  to 
be  stable. 

(2.)  The  supporting  force  may  be  applied  exactly  to  the  cen- 
tre of  gravity.  In  this  case,  if  the  body  be  moved  from  its  ori- 
ginal position,  the  forces  have  their  direction  changed  in  respect 
to  any  line  taken  in  the  body,  and  supposed  to  be  at  rest ;  it  is 
therefore  an  instance  of  that  case  in  parallel  forces,  where  the  forces 
revolve  around  their  points  of  application,  without  ceasing  to  be 
parallel;  for  the  result  will  be  the  same,  whether  the  forces 
themselves  move,  or  the  points  of  application  turn  around  the 
centre  of  force.  Hence,  a  body  supported  by  its  centre  of  gra- 
vity, will  remain  at  rest  in  any  position  in  which  it  is  placed  ; 
the  equilibrium  is  now  said  to  be  indifferent,  as  the  body  has  no 
greater  tendency  to  remain  in  any  one  position  than  in  another. 

(3.)  The  point  of  support  may  be  below  the  centre  of  gravity ; 
in  this  case,  as  the  opposite  directions  of  the  supporting  and 
gravitating  forces  must  coincide  in  the  same  vertical  line,  the 
centre  of  gravity  will  be  immediately  above  the  point  of  sup- 
port. If  the  equilibrium  be  disturbed,  the  centre  of  gravity 
must  describe  a  circle  around  the  point  of  support,  hence  the 
centre  of  gravity  is  in  the  highest  possible  point;  and  as  this 
motion  is  in  a  curve  concave  to  the  horizon,  the  motion  will 
continue  around  the  point  in  the  same  direction  as  at  first, 
until  the  centre  of  gravity  come  immediately  beneath  the  point 
of  support,  or  until  it  meet  some  new  point  of  support,  by 
means  of  which  the  centre  of  gravity  may  be  sustained  in  a  state 
of  stable  equilibrium.  As  the  body  can  never  return  of  itself  to 
its  original  position,  the  equilibrium,  when  it  is  supported  from 
beneath,  is  said  to  be  tottering,  or  unstable. 

All  feats  of  balancing  depend  upon  these  properties  of  the  cen- 
tre of  gravity.  They,  generally  speaking,  consist  in  a  skilful 
application  of  a  small  force  to  retain  the  body  in  its  position  of 
tottering  equilibrium,  even  after  the  conditions  are  partially  dis- 
turbed. Sometimes  the  point  of  support  is  fixed  ;  the  art  then 
consists  in  changing  the  distribution  of  the  weight,  in  such  a  man- 
ner as  to  bring  back  the  line  of  direction  of  the  centre  of  gra- 
vity to  the  point  of  support.  Sometimes  the  point  of  support  is 
moveable,  and  the  skill  is  then  shown,  by  changing  its  position 
in  such  a  manner,  as  to  make  the  line  of  direction  of  the  centre 


104  'OP   THE    CENTRE  \BooklIl. 

of  gravity  constantly  move  its  position,  so  as  to  meet,  and  pass 
through  the  point  of  support. 

107.  A  body  may  be  supported  upon  a  sharp  edge.  In  this  case, 
the  centre  of  gravity  will  be  in  the  vertical  plane  passing  through 
the  edge:  and  here  again  the  equilibrium  may  be  stable,  indiffer- 
ent, or  tottering,  according  as  the  centre  of  gravity  lies  below, 
in,  or  above  the  line,  which  marks  the  meeting  of  the  two  sur- 
faces that  form  the  edge. 

108.  The  body  may  rest  upon  a  surface.    In  this  case,  equili- 
brium can  only  occur  when  the  line  of  direction  of  the  centre  of 
gravity  falls  within  the  base  on  which  the  body  stands.     Al- 
though the  supporting  force  is  a  normal  to  the  surface,  and  in 
order  that  equilibrium  may  exist  theoretically,  the  surface  of  sup- 
port ought  to  be  parallel  to  the  horizon ;  still  there  are  certain 
forces  that  act  to  prevent  a  body  from  sliding,  even  upon  an  in- 
clined surface.     Such  forces  will  be  hereafter  examined  and  de- 
scribed.    It  is  sufficient  for  the  present  to  state,  that  a  body  may 
be  in  equilibrio  upon  a  base  of  small  inclination.     This  however 
can  only  be  the  case,  when,  as  in  the  former  instance,  the  line  of 
direction  of  the  centre  of  gravity  falls  within  the  base. 

109.  If  the  body  be  of  such  a  form  as  to  touch  the  horizontal 
plane,  on  which  it  rests,  only  at  a  single  point,  the  three  several 
species  of  equilibrium  may  exist,  according  to  the  form  of  the 
body,  and  its  position  in  respect  to  the  plane. 

If  from  the  centre  of  gravity  of  the  body,  lines  be  supposed 
to  be  drawn  to  every  point  of  its  surface,  some  of  the  lines  will 
be  always  normals  to  the  surface,  while  others  will  be  oblique. 
If  the  body  rest  upon  any  of  the  points  through  which  one  of 
these  normals  passes,  equilibrium  will  take  place  ;  if  this  normal 
be  the  shortest  line  that  can  be  drawn  from  the  centre  of  gra- 
vity to  the  surface,  the  equilibrium  is  stable,  for  the  centre  of 
gravity  will  be  directly  above  the  point  of  support,  and  will  also 
be  in  the  lowest  possible  position;  hence  any  disturbing  force 
will  only  cause  an  oscillation  in  the  body,  which  will  finally  re- 
turn to  rest  in  its  original  position.  The  same  is  the  case  if  the 
normal,  although  not  absolutely  the  shortest  line  drawn  from  the 
centre  of  gravity  to  the  surface,  is  relatively  shorter  than  those 
contiguous  to  it.  If  the  body  rest  upon  the  point  where  the 
longest  normal  intersects  the  surface,  the  body  rs  in  a  state  of  tot- 
tering equilibrium,  and  will,  if  disturbed,  turn  around  until  it 
rest  upon  the  shortest  normal ;  the  same  will  occur,  even  if  the 
normal  be  not  absolutely  the  longest  line,  but  if  it  be  longer  than 
the  other  lines,  drawn  from  the  centre  of  gravity  to  the  surface, 
which  are  in  its  immediate  vicinity.  If  all  the  lines  drawn  from 


Book  IfJ.] 


OF    GRAVlfY. 


105 


the  centre  to  the  surface  be  normals,  the  equilibrium  will  be  in- 
different, or  if  the  normal  on  which  it  rests  be  situated  in  the 
vicinity  of  other  lines  that  are  also  normals. 

As  instances  : 

A  portion  of  a  homogeneous  sphere,  or  of  a  spherical  surface 
equal  to,  or  less  than,  a  hemisphere,  will  have  stable  equilibrium; 
the  same  will  take  place  in  an  ellipsoid  formed  by  the  revolution 
of  an  ellipse  around  its  shorter  axis,  which  will  come  to  rest  with 
that  axis  in  a  vertical  position. 

A  homogeneous  sphere  will  remain  in  any  position  in  which  it 
is  placed  upon  a  plane,  and  is  hence  in  a  state  of  indifference. 

An  egg,  or  an  oblong  ellipsoid  of  revolution,  will  be  in  a  state 
of  tottering  equilibrium,  if  poised  upon  its  longer  axis  ;  while  if 
laid  on  one  side,  as  all  the  radii  of  the  circular  section,  in  which 
the  points  of  contact  are  situated,  pass  through  the  centre  of  gra- 
vity, the  equilibrium  is  indifferent. 

A  portion  of  a  cylinder  not  greater  than  the  half,  cut  off  by  a 
plane  parallel  to  its  axis,  and  laid  on  the  curved  surface,  comes  to 
restupon  the  line,  in  which  a  plane  perpendicular  to  that  by  which 
it  is  cut  from  the  cylinder  intersects  the  surface,  and  is  there- 
fore in  a  state  of  stable  equilibrium. 

These  circumstances  may  be  illustrated  by  the  following 
figures. 

FIG.  1  FIG.  2. 


Let  the  above  figures  represent  bodies  whose  sections  are  se- 
micircular, Fig.  1  being  solid,  Fig.  2  being  hollow.  In  either 
case,  the  normal  #D  will  be  the  shortest  line  that  can  be  drawn 
from  the  centre  of  gravity,  g-,  to  the  curved  surface.  If  any  dis- 
turbing force  act,  that  is  not  sufficient  to  bring  the  body  into  the 
position,  in  which  lines  passing  through  #,  and  A  or  B,  are  verti- 
cal, the  body  will  finally  return  to  rest  in  the  position  in  which 
g-D  is  vertical :  when  the  disturbing  force  is  removed,  it  will  os- 
cillate in  returning  to  rest,  until  the  resistances  overcome  its  mo- 
tion. In  the  case  of  a  spherical  surface,  the  oscillations  may  take 
place  in  any  direction  whatsoever,  but  in  the  case  of  a  portion  of 
a  cylinder,  only  in  planes  parallel  to  the  circular  section.  In  a 
solid  spheric  segment,  the  distance  Cg  is  only  fths  of  CD ;  while 
in  a  hollow  spheric  segment,  the  centre  of  gravity  will  coincide 

14 


106 


OP    THE    CENTRE 


[So ok  HI 


with  that  of  the  curve,  and  its  distance  from  C  will  be  nearly 
fds  of  CD.  A  similar  difference  in  position  will  take  place,  in 
relation  to  portions  of  solid,  and  of  hollow  cylinders.  Hollow 
bodies  of  these  classes,  are,  therefore,  more  stable  than  solid  ones  ; 
and  with  equal  weights,  will  more  powerfully  resist  any  effort  to 
disturb  their  equilibrium. 

Jn  the  homogeneous  sphere,  one  of  whose  great  circles  is  re- 
presented  by  the  circle  ADBE,  Fig.  1,  the  centre  of  gravity, 
FIG.  1.  FIG,  2. 


corresponds  with  the  centre  of  magnitude,  and  the  lines  of  direc- 
tion will  be  all  normals,  and  of  equal  length,  upon  whatever 
point  it  rest;  hence  its  equilibrium  is  that  of  indifference.  If  the 
figure  represent  the  circular  section  of  a  cylinder,  a  similar  state 
of  indifference  will  exist  in  one  direction.  If  in  Fig.  2,  the 
elliptical  curve  ADBE,  represent  the  section  of  an  ellipsoid, 
formed  by  the  revolution  of  the  curve  upon  its  shorter  axis,  and 
the  solid  thus  formed  be  homogeneous,  the  centre  of  gravity  will 
be  in  the  centre  of  magnitude  g.  The  lines  of  direction  #D  and 
#E,  which  are  normals  to  the  surface,  are  the  shortest  that  can  be 
drawn  within  the  body,  it  will  therefore  have,  when  resting  upon 
ci'.her  of  the  points  D  or  E,  a  stale  of  stable  equilibrium.  All 
lines  in  the  plane  of  #A,  and#B,  are  also  normals  to  the  surface, 
but  are  the  longest  lines  of  direction  that  can  be  drawn  within 
the  body  :  hence,  if  resting  upon  the  points  A  or  B,  it  will  be  in  a 
state  of  tottering  equilibrium,  in  case  iho  disturbing  force  act  in 
the  plane  of  AB.  If  the  same  curve  represent  the  section  of  an 
ellipsoid,  formed  by  revolution  around  its  longer  axis  AB,  the 
lines  g\  and  #B,  are  normals,  and  the  two  longest  lines  of  direc- 
tion that  can  be  drawn  within  the  body,  resting  on  the  points  A 
and  B  ;  therefore,  its  equilibrium  is  tottering.  All  the  lines  of  di- 
rection that  can  be  drawn  in  the  plane  of  DE,  are  equal  among 
themselves,  and  shorter  than  any  other  lines  that  can  be  drawn 
within  the  body  from  the  point  g  ;  hence  in  respect  to  forces  act- 
ing in  the  plane  of  DE,  the  equilibrium  is  indifferent. 

110.  If  the  surface  of  a  body  resting  on  a  plane,  be  also  a 
plane,  and  the  line  of  direction  of  the  centre  of  gravity  fall 
within  it,  the  equilibrium  is  of  course  stable.  If  extrinsic  forces 
act  to  disturb  the  position  of  the  body,  the  more  extensive  the 


Book  ///.] 


OF    GRAVITY. 


107 


plane  surface  on  which  it  rests,  the  nearer  to  the  centre  of  the 
surface  the  line  of  direction  falls,  and  the  lower  the  position  of 
the  centre  of  gravity,  the  more  the  body  will  resist  a  force  ap- 
plied to  overturn  it.  When  the  force  that  acts  to  overturn  it  is 
sufficient  for  the  purpose,  the  body,  if  not  broken  by  its  action, 
will  turn  around  one  of  its  solid  angles  as  a  centre,  or  round  one 
of  its  edges  as  an  axis.  The  centre  of  gravity  must  of  course 
rise  in  a  circular  arc,  and  with  it  the  weight  of  the  body  ;  the 
resistance  of  the  body  to  the  effort  to  overturn  it,  will  therefore 
depend,  not  only  upon  its  own  weight,  but  upon  the  position  and 
curvature  of  the  arc  described  by  the  centre  of  gravity.  Some 
of  the  cases  that  may  occur  in  practice,  are  represented  below. 

The  triangle  ABC,  Fig.  1,  is   the  section  of  a  pyramid   or 
cone  whose  centre  of  gravity,  g,  is  at  the  distance  of  fths  of  its 

FIG.  1.  FIG.  2. 


height  from  the  vertex.     In  overturning,  its  centre  of  gravity 

would  describe  a  circular  arc  around  the  corner  C.     In  Fig.  2t 

the  triangle  represents  the  section  of  a  triangular  prism,  whose 

centre  of  gravity  is  in  the  point  g,  at  a  distance  of  fds  of  its 

height  from  the  vertex.     The  weight  in  the  former  case  will  act 

more  directly  to  preserve  the  stability,  while  the  disturbing  force 

will  act  more  obliquely.     The  former  is  therefore  the  most  stable. 

In    the  figures  beneath,  Figs.    1   and  2,  respectively   repre- 

FIG.  1.  FIG  2. 


B    C 


sent  sections  of  prisms,  the  first  of  which  has  for  its  section  a 
trapezium,  with  parallel  bases;  the  second  is  rectangular.  The 
centre  of  gravity  of  the  second  is  at  half  its  height,  of  the  first, 
at  a  distance  from  CD,  represented  by  the  formula 

_o_      c— 2d 
A~  3  '    c— d 


108 


OF    THE    CENTRE 


[Book  III. 


c  being  the  greater  and  d  the  lesser  base  ;  it  is  obvious  then  that 
its  centre  of  gravity  will  be  lower  than  in  Fig.  2,  and  it  will  in 
consequence  be  more  stable. 

Had  Fig.  1  been  the  section  of  a  truncated  pyramid,  the  centre 
of  gravity  would  have  been  still  nearer  the  base,  and  the  stability 
greater.  Did  Fig.  1  rest  on  the  base  AB,  the  results  would  be 
directly  opposite. 

In  two  rectangular  prisms,  whose  sections  are  Fig.  I  and  2, 
the  stability  of  the  lowest,  Fig.  2,  is  the  greater  of  the  two, 
FIG.  1.  FIG.  2. 


their  bases  being  equal ;  and  in  two  prisms  of  equal  height,  but 
of  different  bases,  that  with  the  greatest  base  will  have  the  great- 
est stability. 

If  however  the  prism,  Fig.  1,  should  cease  to  be  homogeneous, 
and  be  loaded  with  a  weight  towards  the  base  CF,  h}^  means  of 
which  its  centre  of  gravity  is  lowered  to  #',  whose  distance  from 
the  plane  of  support  is  equal  to  g  in  Fig.  2,  the  two  bodies  will 
have  equal  degrees  of  stability. 

It  may  hence  be  inferred,  that  pyramids  and  cones,  of  small  al- 
titude, compared  with  theextent  of  their  bases,  are  among  the  most 
stable  of  all  geometric  figures.  That  walls  with  a  broad  base,  and 
whose  faces  incline  inwards,  are  more  stable  than  those  whose 
surfaces  are  parallel;  that  with  equal  bases,  walls  of  the  least 
heights,  and  with  equal  heights,  those  with  the  greatest  bases  are 
the  most  stable;  that  stability  may  be  given  to  bodies,  by  con- 
structing them  in  such  a  manner  that  their  centre  of  gravity 
may  fall  below  the  point  in  which  it  would  be  if  they  were  ho- 
mogeneous. 

111.  A  body,  whose  sides  are  inclined  in  such  a  way  that  it 
overhangs  on  one  side  of  the  base,  may,  notwithstanding;,  be  sta- 
ble, if  the  centre  of  gravity  fall  within  the  base.  And  even  if 
the  vertical  line  that  passes  through  its  centre  of  magnitude  fall 
without  the  base,  the  actual  centre  of  gravity  may  be  so  lowered, 
by  a  proper  distribution  of  the  weight,  that  its  line  of  direction 
shall  fall  within  the  base,  and  stability  ensue.  Thus  in  the  city 
of  Pisa,  in  Italy,  there  is  a  tower  that  leans  so  much  to  one  side, 


Book  III.}  OF    GRAVITY.  109 

that  it  not  only  appears  unsafe,  but  would  be  actually  so,  if  ho- 
mogeneous. Rut  by  an  ingenious  arrangement  of  the  materials, 
it  is  rendered  stable.  The  lower  parts  are  built  of  a  heavy  vol- 
canic rock  ;  the  middle  of  brick  ;  and  the  top  of  a  light  porous 
stone,  that  will  float  on  water  :  hence,  the  centre  of  gravity  is 
so  low,  that  its  line  of  direction  falls  within  the  base. 

1 12.  It  is  not  necessary  that  the  base  shall  be  actually  a  plane 
surface;  but  it  is  sufficient  that  the  body  rest  upon  points.     If  it 
rest  upon  no  more  than  two  points,  it  is  in  the  condition  of  a  body 
resting  upon  an  edge,  and  the  line  of  direction  of  the  centre  of 
gravity  must  fall  in  the  line  that  joins  these  points,  otherwise  the 
body  will  not  be  in  equilibrio.     If  itrest  on  more  than  two  points, 
the  base  is  the  surface  formed  by  joining  the  points  by  straight 
lines.     It  may  in  like  manner  rest  on  two  edges,  and  the  base 
will  be  defined  by  supposing  their  extremities  to  be  joined.     If 
the  surface  on  which  the  body  rests  be  irregular,  it  is  best  sup- 
ported upon  three  points  ;  for  these  lie  always  in  one  plane,  and 
the  stale  of  the  body  is  precisely  the  same  as  if  this  plane  were 
applied  to  another.     This  principle  is  applied  in  practice  to  a 
variety  of  surveying  and  astronomic  instruments,  to  which  sta- 
bility is  given  by  placing  them  upon  three  feet;  if  they  had  more 
than  three  feet,  they  would  rest  firmly  upon  no  surface  but  one 
perfectly  plane,  or  at  least  having  in  it  an  equal  number  of  points, 
on  which  to  place  the  feet,  that  lie  in  one  plane  :  while  with  three 
feet,  they  can  be  placed  steadily  on  the  most  irregular  surfaces. 

A  three-legged  table  or  chair  stands  firmly  on  the  most  un- 
equal floor,  while  one  with  four  legs  is  unsteady,  except  upon  a 
floor  that  is  perfectly  level. 

1 13.  We  have  stated  that  there  are  cases  in  nature,  in  which  a 
body  will  not  descend  an  inclined  plane.     In  such  a  case,  if  the 
line  of  direction  fall  within  the  base,  the  body  will  be  stable  in 
spite  of  the  inclination  of  the  plane  ;  and  if  it  descend  upon  the 
plane,  it  will  slide  down  it. 

Thus  in  the  body  beneath,  if  the  centre  of  gravity  be  at  g, 
its  line  of  direction  falls  within  the  base  ;  if  the  plane  oppose  a 
resistance  to  its  descent,  it  is  stable  ;  but  if  the  plane  oppose  no 
resistance,  it  will  slide  down. 

If  a  body  beplaced  onan  inclined  plane, 
and  the  line  of  direction  of  the  centre  of 
gravity,  fall  without  the  base,  it  will 
turn  around,  until  it  apply  itself  to  the 
plane  by  a  surface,  within  which  its  line 
of  direction  will  fall. 


110 


OF    THE    CENTRE 


[Eook  III. 


Thus  the  body  whose  section  is  represented  beneath  at  A,  will 
be  overturned,  and  come  into  the  position  B  ;  it  will  there  re- 
main at  rest,  if  supported  by  a  resistance  in  the  plane,  or  will 
slide  down  it  if  not  supported. 

If  the  body  have  no  surface 
wilhin  which  its  line  of  direction 
can  fall,  it  will  roll  down  the 
inclined  plane  ;  thus  the  body 
whose  section,  represented  be- 
neath, is  a  regular  polygon,  will 
roll  down  the  inclined  plane  on 
_  which  it  rests. 

A  cylinder  or  sphere,  having  a  circular  sec- 
tion, will  not  rest,  if  homogeneous,  on  an  in- 
clined plane,  at  any  of  its  points.  But  it  may,  if 
loaded  by  a  weight  which  will  cause  the  line  of 
direction  to  fall  upon,  or  above  the  place  where 
it  rests  upon  the  plane,  either  remain  at  rest, 
or  actually  move  up  the  inclined  plane. 

Thus  in  the  body  whose  circular  section  is 
represented  beneath,  an  eccentric  weight  at 
W  will  change  the  position  of  the  centre  of  gravity  from  c  to  g, 
and  the  body  will  roll  up  the  plane,  until  the  line  of  direction 
fall  upon  the  point  at  which  it  meets  the  plane,  where  it  will 
come  to  rest. 


If  this  weight  be  moved,  as  may  be  done  by  a  spring,  in  such 
a  manner  as  to  be  raised  as  much  as  it  tends  to  fall  by  the  rolling 
motion  of  the  body,  the  latter  will  exhibit  the  curious  pheno- 
menon, of  a  body  apparently  mounting  in  opposition  to  gravity. 

A  double  cone  may  be  made  to  appear  to  roll  upwards,  by 
placing  it  between  two  edges,  inclined  to  the  horizon,  and  to  each 
other,  and  meeting  at  an  acute  angle  at  their  lowest  points.  If 
the  double  cone  be  laid  at  this  angle,  it  rests  upon  its  greatest 
section  ;  and  if  the  inclination  of  the  planes  to  the  horizon  be 
such,  that  the  cone,  when  laid  at  other  points  of  the  plane,  shall 


Book  HI}  OP    INERTIA.  Ill 

rest  upon  two  of  its  sections  whose  radius  has  lessened  more  than 
the  height  of  the  plane  has  increased,  the  cone  if  laid  at  the 
place  where  the  planes  meet,  will  roll  along  them,  appearing  to 
ascend  them,  when  in  fact  its  centre  of  gravity  is  constantly  de- 
scending. 

The  distinguishing  property  of  the  centre  of  gravity  in  a 
solid  body,  is,  that  if  it  be  supported,  the  body  is  supported  ;  but 
if  it  be  not  supported,  the  body  will  fall,  and  continue  to  fall,  un- 
til it  meet  a  resistance  of  such  a  nature  as  to  support  this  point. 

114.  In  irregular  bodies,  whether  the  irregularity  arise  from 
mere  figure,  or  from  an  unequal  distribution  of  matter  throughout 
their  bulk,  the  mathematical  methods  of  finding  the  position  of  the 
centre  of  gravity  are  inapplicable.     We  may  in  such  cases  have 
recourse  to  experimental  methods,  whose  principles  are  founded 
on  the  properties  of  the  centre  of  gravity. 

(1.)  The  body  may  be  suspended  alternately,  from  two  differ- 
ent points  in  its  surface.  The  centre  of  gravity  will,  in  either 
case,  lie  immediately  beneath  the  point  of  suspension  ;  it  will  there- 
fore lie  at  the  common  intersection  of  the  two  lines  that  join  the 
two  points  of  suspension  to  points  in  the  body  situated  directly 
benea,th  each  of  them,  when  it  is  suspended  from  it. 

(2  )  The  body  may  be  made  to  rest  in  equilibrio,  in  three  dif- 
ferent positions,  upon  a  sharp  edge;  the  vertical  plane  passing 
through  the  edge,  in  each  of  the  three  several  positions  of  the  body, 
will  also  pass  through  the  centre  of  gravity,  and  the  common 
intersection  of  the  three  planes  determines  the  situation  of  this 
point. 

115.  If  a  body  move  in  a  straight  line,  under  the  action  of  any 
other  force  than  that  of  gravity,  each  of  its  particles  may  be  sup- 
posed to  be  actuated  by  an  equal  and  parallel  force;   hence  it  will 
act  as  if  its  whole  mass  were  collected  in  the  centre  of  these  pa- 
rallel forces.     This  point,  which  in  gravitating  bodies,  as  has  just 
been  seen,  is  called  the  centre  of  gravity,  is,  in  this  case,  called 
the  Centre  of  Inertia.     Its  position  may  therefore  be  found  by 
the  same  processes,  whether  analytic,  geometric,  or  experimental, 
by  which  the  centre  of  gravity  can  be  found. 


11)8  OF    FKICTION.  [Book  HI. 

CHAPTER  V. 

OF  FRICTION. 

• 

116.  So  far  as  our  investigations  have  hitherto  proceeded,  it 
might  appear,  that  so  soon  as  equilibrium  ceases  to  exist,  among 
the  forces  that  act  upon  a  body,  it  must  be  set  in  motion.     This, 
however,  does  not  take  place  in  practice  ;  for  there  are  resist- 
ances that  are  incapable  themselves  of  causing  motion,  and  which 
therefore  do  not  come  within  our  original  definition  of  the  term 
force  ;  these  are  yet  effectual  in  retaining  bodies  at  rest,  after  the 
theoretic  conditions  of  equilibrium  are   at  end;  they  are,  also, 
capable  of  bringing  bodies  to  rest,   when  they   have  been  pre- 
viously set  in    motion.     Thus   then,   although  they  do  not  fall 
within  the  strict  definition  of  forces,  still  we  cannot  determine 
the  circumstances  under  which  bodies  move,  without  taking  them 
into  account.     The  retardation  they  produce  in  motions  arising 
from  other  forces,   is  capable  of  being  estimated  in  terms  of  a 
conventional  unit,  precisely  as  if  they  were  accelerating  forces, 
acting  in  directions  opposed  to  those  of  the  motion  due  to  other 
forces.     Hence  we  may  consider  the  action  of  these  resistances, 
precisely  as  if  they  were  forces,   always  acting  in  directions  op- 
posite to  those  of  a  previously  communicated  motion,  or  to  lhat 
in  which  a  body  would  tend  to  move,  when  its  equilibrium  is  dis- 
turbed.    They  are,  in  fact,  passive  or  resisting  forces,  that  are 
only  called  into  action  under  certain  circumstances,  but  which 
have,  like  active  forces,  a  determinate  measure,  a  definite  inten- 
sity, and  a  known  point  of  application. 

Of  such  resisting  or  retarding  forces,  the  more  important 
are  : 

The  resistance  that  the  surfaces  of  solid  bodies  oppose  to  each 
other's  motions,  or  that  one  opposes  to  the  motion  of  the  other. 
This  is  called  Friction. 

The  resistance  that  certain  bodies  oppose  to  flexure; 

The  resistance  of  fluid  media  to  bodies  moving  in  them,  and 
which  solids  oppose  to  the  motion  of  fluids. 

The  two  first  of  these  are  of  direct  importance  to  our  present 
subject,  and  may  be  examined  by  the  aid  of  principles  that  have 
already  been  laid  down.  The  consideration  of  the  third  must  ne- 
cessarily be  postponed,  until  we  treat  of  the  mechanics  of  fluid 
bodies. 

117.  The  precise  nature  of  friction  is  unknown  to  us,  although 
there  is  a  well-founded  hypothesis  on  the  subject  that  shall  here- 


III.}  OF    FRICTION.  113 

after  beeited.  We  are  therefore  compelled  to  have  recourse  to 
experiment,  in  order  to  ascertain  the  laws  its  action  follows.  The 
more  important  of  these  experiments,  so  far  as  they  have  been 
recorded,  are  those  of  Coulomb,  Vince,  and  Ximenes.  To  these 
we  shall  recur,  describing  the  manner  in  which  they  were  per- 
formed. 

US.  Friction,  although  always  arising  from  the  same  general 
cause,  may  be  classed  into  three  distinct  varieties  : 

(1.)  That  which  occurs  when  one  body  slides  upon  the  surface 
of  another ; 

(2.)  The  friction  of  bodies  rolling;  and 

(3.)  The  friction  at  the  axles  of  wheels. 

These  have  all  been  separately  and  fully  examined,  and  we 
shall  now  proceed  to  describe  the  manner  in  which  the  various 
experiments  were  made,  and  to  indicate  the  result  that  have 
been  obtained  from  them. 

119.  If  a  body  be  placed  upon  a  horizontal  plane,  and  the  line 
of  direction  of  its  centre  of  gravity  fall  within  its  base,  it  will  be 
at  rest,  under  two  countervailing  forces,  its  weight,  and  the 
resistance  of  the  plane.  If  the  plane  be  gradually  inclined,  al- 
though the  equilibrium  of  these  two  forces  is  disturbed,  because 
they  no  longer  act  in  direct  opposition  to  each  other,  the  body 
will  not  at  first  move,  but  will  remain  at  rest  until  the  plane  ac- 
quire an  inclination  ;  this  inclination  will  be  different,  according 
to  the  nature  of  the  surface  and  figure  of  the  body.  At  this  in- 
clination the  body  will  be  set  in  motion,  and  will  slide  or  roll 
down  the  inclined  plane,  according  to  the  manner  the  line  of  di- 
rection falls.  Up  to  the  beginning  of  its  motion,  it  is  supported 
by  its  friction  upon  the  plane,  and  this  friction  will  be  represented 
in  direction,  by  a  force  parallel  to  the  plane  on  which  it  moves. 
At  the  instant  before  motion  begins,  the  three  forces,  namely, 
the  weight,  the  pressure,  and  the  friction,  are  exactly  in  equili- 
brio;  their  respective  intensities  may  therefore  be  represented 
upon  the  principles  in  §  15. 

Supposing  the  weight,  W,  to  be  known,  and  the  angle  *  of  the 
plane's  inclination  to  the  horizon  to  be  determined,  the  value  of 
the  friction,  F,  and  pressure,  P,  will  be 

F=W  sin.  «,     P=W  cos.  i ; 

and  the  friction  will  be  given  in  terms  of  the  pressure,  by  the  ex- 
pression 

F— P  tan.  t. 

Experiments  conducted  upon  this  principle,  have  given  the 
following  results  : 

15 


1 14  OF  FRICTION.  [Book  ///. 

(1.)  The  friction  is  greatest  between  rough  surfaces,  and  di- 
minishes with  the  degree  of  polish  that  is  given  to  them. 

(2.)  Friction  is  greater,  all  other  things  being  equal,  between 
the  surfaces  of  bodies  that  are  of  the  same  material,  or  homo- 
geneous, than  it  is  between  bodies  of  different  materials. 

(3.)  The  rubbing  surfaces  remaining  the  same,  the  friction  is 
directly  proportioned  to  the  pressure. 

(4.)  The  friction  does  not  increase  or  diminish  with  the  area 
of  the  rubbing  surface,  the  weight  and  the  nature  of  the  surface 
remaining  the  same. 

These  experiments  are  limited,  by  their  very  nature,  to  the 
determination  of  the  resistance  that  prevents  a  body  from  being 
set  in  motion  ;  and  it  might,  at  first  sight,  appear  more  than  pro- 
bable, that  this  force  is  more  intense  than  the  friction  which  re- 
tards the  velocity  of  a  moving  body.  Neither  do  they  give  any 
information  whether  the  intensity  of  friction  bears  any  relation 
to  the  velocity.  Another  defect  arises  from  the  small  num- 
ber of  the  experiments  that  have  been  performed  in  this  manner, 
and  we  are  hence  uncertain,  whether  the  results  are  applicable  to 
all  cases  whatsoever,  or  limited  to  a  few  particular  instances. 

120.  The  experiments  of  Coulomb,  Vince,  and  Ximenes,  were 
performed  in  another  manner.  A  body  was  drawn  along  a  hori- 
zontal table,  by  means  of  a  weight  attached  to  it  by  a  cord,  and 
this  cord  passed  over  a  pulley.  The  weight  that  produces  a  con- 
stant velocity  is  obviously  the  measure  of  the  friction,  which  is, 
in  this  case,  the  resistance  that  opposes  the  motion  of  bodies  ; 
the  weight  necessary  to  set  them  in  motion,  is  the  measure  of  the 
resistance  that  opposes  their  passage  from  a  state  of  rest. 

The  different  circumstances  to  be  examined  are : 

(1.)  The  relation  of  the  friction  to  the  pressure  ; 

(2.)  The  effect  of  the  nature  of  the  surfaces,  and  of  the  manner 
of  their  preparation,  upon  the  friction; 

(3.)  The  variation  in  the  friction  produced  by  a  longer  or 
shorter  continuance  of  the  contact,  previous  to  the  application 
of  a  force  to  overcome  it ; 

(4.)  The  influence  of  the  extent  of  the  surface  ; 

(5.)  The  change,  if  any,  at  different  velocities. 

(I.)  As  respects  the  relation  of  the  friction  to  the  pressure. 

When  at  a  maximum,  all  other  circumstances  remaining  the 
same,  the  friction  was  found  to  have  a  constant  relation  to  tho 
pressure.  This  maximum,  as  will  hereafter  be  seen,  is  that  which 
opposes  the  force  that  acts  to  set  a  body  in  motion. 

The  friction  of  surfaces,  that  had  been  at  rest  until  the  maxi- 
mum was  attained,  was  found  to  be  as  follows,  taking  the  mean 
of  the  experiments  : 


Book  III.]  OF    FRICTION.  115 

^ 

3 


Of  Oak  resting  upon  Oak,  the  fibres  being  parallel,     -     -     - 


Of  Oak  resting  upon  Oak,  the  fibres  being  at  right  angles  to  1  1 

each  other,       - )         3.75 

Of  Oak  resting  upon  Fir,    ------------      

Of  Fir  resting  upon  Fir, 

1.78 

Of  Iron  resting  upon  Oak,        -----------      

5.5 

Of  Iron  resting  upon  Iron,       fi.- *s  is.     -     -     -     ...     -     -      

3*5 

Of  Iron  resting  upon  Brass,      --- --      

3.8 

and  if  well  polished,  and  the  surfaces  small,    ------          _ 

6 

The  friction  of  bodies  in  motion  was  found  considerably  less, 
being  at  a  mean  as  follows  : 

Of  Oak  moving  on  Oak, -----      

9.5 

Oak  moving  on  Fir,        _-_._-------.      

6.3 

Fir  moving  on  Fir, --------- 

6 

Elm  moving  on  Elm, >----        — 

Iron  or  Copper  moving  on  wood,        --_--._._        — 

13 

Iron  on  Iron,         -_.---------_-_    

3.55 

Iron  on  Copper,  after  long  attrition, -          _ 

6 

(2.)  It  was  found  that  the  metals  could  be  polished  separately, 
until  the  minimum  of  friction  was  attained,  but  that  in  soft  sub- 
stances, it  was  necessary  that  they  should  be  compressed,  either 
in  the  course  of  the  experiments,  or  by  previous  pressure.  Thus 
i-n  newly  planed  wood,  the  maximum  friction  was  not  far  from 
equal  to  the  pressure,  while  after  being  some  time  in  use,  it  fell 
to  the  ratio  at  which  it  has  been  slated,  of  i. 

The  polishing  of  the  surfaces  of  iron  and  copper,  rubbing  against 
each  other,  reduced  the  friction  nearly  one  half. 

Unctuous  substances,  interposed  between  the  rubbing  surfaces, 
diminish  the  friction  ;  and  the  experiments  of  Coulomb  showed, 
that  those  which  are  hardest  have  the  greatest  effect,  when  the 


116  OF    FRICTION.  [Book  Iff. 

weight  is  great;  thus,  although  in  light  and  delicate  machinery, 
the  most  limpid  oils  are  hest,  they  are  wholly  ineffectual  in  great 
pressures,  when  tallow  must  be  employed.  At  still  higher  pres- 
sures than  those  which  became  the  subject  of  experiment,  it  ap- 
pears from  practical  observations,  that  tcllow  loses  its  power  of 
diminishing  friction.  So  also  is  it  less  effective  when  the  velo- 
locity  becomes  sufficient  to  melt  it.  In  this  case  it  acts  as  if  it 
were  oil,  being  well  suited  to  diminish  friction  when  the  weight 
is  light,  but  of  little  effect  in  great  pressures.  When  oleaginous 
substances  cease  to  have  effect,  as  tallow,  when  the  pressure  be- 
comes too  intense,  or  the  heat  produced  by  the  velocity  is  effi- 
cient te  render  it  fluid,  solids  of  an  unctuous  texture  have  been 
found  to  answer  the  purpose.  Thus  in  carriages  moving  rapidly, 
and  in  large  machinery,  plumbago  has  been  mixed  with  tallow: 
it  has  also  been  used  separately ;  and  recently,  soap-stone 
(steatite)  has  been  found  efficacious  in  lessening  friction  when 
the  weight  was  so  great  that  all  other  means  failed. 

Within  the  limits  of  the  experiments,  the  mean  friction,  when 
oleaginous  matters  were  interposed,  was  found  to  be  as  follows  : 

Tallow  interposed  between  two  surfaces  of  oak, 

If  wiped  off  after  application, ...._        

In  very  small  surfaces,        -----„_----.        

Tallow  interposed  between  wood  and  oak,  moving  slowly  on   )  1 

each  other,       -.- .     .     .     .     _    j  35 

Between  brass  and  oak,  under  similar  circumstances,        -     -     - 

47 
After  being  sometime  in  use  the  friction  was  increased  to  )        1        ,1 

thrice  these  amounts,  or  to         --.--..     j      12 &      16 
The  same  result  took  place  at  increased  velocities. 
In  the  case  of  tallow,  interposed  between  metallic  surfaces,  the 

frictions  were  the  same  at  all  velocities. 

Tallow,  interposed  between  two  surfaces  of  iron,  gave       -     -     -        _L 

10 

Between  iron  and  copper,         -     -     -     -     -     -     -     -     .^i    ;.        JL 

While 

Oil  interposed  between  the  same  surfaces,  gave  for  the  friction, 

8 

(3.)  It  was  found,  that  in  the  case  of  the  metals  resting  upon 
each  other,  the  maximum-of  friction  was  reached  instantly;  in 
wood  resting  upon  wood,  the  maximum  was  reached  in  a  few 
in  the  contact  of  metals  with  wood,  the  maximum  was 


Book  HI.]  OF    FRICTION.  l  ^ 

not  attained  until  some  days  had  elapsed  ;  and  when  grease  was 
interposed  between  any  substances  whatsoever,  the  time  during 
which  the  friction  continued  to  increase,  was  still  longer  than  in 
the  latter  case. 

The  relations  that  have  been  stated  in  the  two  first  instances, 
were  not  found  to  be  wholly  independent  of  the  amount  of  pres- 
sure. That  is  to  say,  the  ratio  of  the  surface  to  the  pressure  was 
not  constant,  but  appeared  to  diminish  at  the  higher  pressures, 
that  were  the  objects  of  experiment.  This  diminution  was  even 
more  obvious,  in  the  experiments  ofVince  and  Ximenes,  than  in 
those  of  Coulomb. 

(4.)  The  magnitude  of  the  surface  was  found  to  have  an  appre- 
ciable effect  upon  the  friction.  In  the  case  of  oak  moving  upon 
oak,  in  pressures  from  100  to  4000  Ibs.  per  square  foot,  an  adhe- 
sion or  additional  resistance  was  found  amounting  to  about  If  Ibs. 
per  square  foot ;  and  in  all  other  cases,  an  analogous  increase  was 
remarked. 

(5.)  Under  similar  circumstances,  the  weight,  that  overcame 
the  friction,  was  found  to  be  nearly  the  same  at  all  velocities. 
The  slight  variations  that  were  observed,  seemed  rather  to  lead 
to  the  conclusion,  that  at  small  velocities  the  friction  increases 
with  the  velocity,  but  that  at  great  velocities,  it  diminishes  in 
the  ratio  of  some  small  function  of  the  velocity. 

It  thus  appears  that  the  variation  in  pressure,  in  magnitude  of 
surface,  and  in  velocity,  have  but  little  effect,  unless  they  become 
very  great. 

We  may  therefore,  in  all  usual  cases,  assume  the  following  laws 
as  sufficiently  accurate  for  practical  purposes. 

(1.)  Between  similar  substances,  under  similar  circumstances, 
Friction  is  a  constant  retarding  force. 

(2.  )  Friction  is  greatest  between  bodies  whose  surfaces  are 
rough,  and  is  lessened  by  polishing  them. 

(3.)  It  is  greater  between  surfaces  composed  of  the  same  ma- 
terial, than  between  bodies  composed  of  different  materials. 

(4.)  If  the  rubbing  surfaces  remain  the  same,  the  friction  in- 
creases directly  as  the  pressure. 

(5.)  If  the  pressure  continue  the  same,  the  friction  has  no  re- 
lation to  the  magnitude  of  the  surface. 

(6.)  The  relation  between  the  pressure  and  the  friction,  can- 
not be  safely  taken  at  less  than  1,  in  calculating  the  effect  of  prime 
movers. 

On  the  other  hand,  when  friction  is  to  be  substituted  for  a  fixed 
resistance,  it  cannot  safely  be  estimated  at  more  than  ^. 

Although  the  weight,  which  measures  the  intensity'of  friction, 
be  constant,  whatever  be  the  velocity,  still  as  the  measure  of  a 
force  is  not  merely  the  weight  that  it  is  capable  of  raising,  but  de- 


118  or  FRICTION.  [Book  HI. 

pends  also  upon  the  rate  at  which  the  weight  is  lifted,  the  power 
that  is  applied  to  overcome  the  constant  friction,  must  increase 
with  the  velocity  at  which  that  resistance  is  overcome. 

The  several  laws  that  have  just  been  laid  down,  are,  as  is  ob- 
vious from  the  result  of  the  experiments,  not  absolutely,  but  only 
nearly  true,  and  cases  occur  occasionally  in  practice,  in  which 
the  minute  effects  that  are  due  to  the  increase  of  pressure  of  the 
surface,  and  of  the  velocity,  become  important.  Thus  :  in  launch- 
ing a  ship,  the  vast  weight  which  it  has,  appears  to  become  a  force 
capable  of  very  much  lessening  the  friction,  and  the  vessel  de- 
scends down  a  plane  of  less  inclination  than  a  lighter  body  would. 
At  the  Shoot  of  Alpnach,  in  Switzerland,  where  trees  of  great 
size  are  conveyed  along  a  trough  of  but  small  inclination,  for  7 
or  8  miles,  in  consequence  of  a  velocity  previously  acquired  in 
falling  through  a  curved  spout  of  great  inclination,  the  phenome- 
na, as  noted  by  Playfair,  are  such  as  can  only  be  explained  by  as- 
suming, that  the  friction  of  great  masses,  moving  with  great  ve- 
locities, is  considerably  less  than  that  of  smaller  bodies  moving 
slowly. 

121.  The  friction  of  rolling  bodies  has  also  been  investigated 
experimentally  by  Coulomb.  The  following  are  deductions  from 
his  experiments  : 

(1.)   Like  the  friction  of  sliding  bodies,  it  is  a  constant  force; 

(2.)  It  is  affected  by  the  nature  of  the  surface,  so  far  as  polish 
is  concerned,  but  is  not  lessened  by  the  interposition  of  oleagi- 
nous and  unctuous  substances; 

(3  )  It  is  less  between  heterogeneous  than  between  homogen- 
eous substances  ; 

(4.)  It  is  directly  proportioned  to  the  pressure  ; 

(5.)   It  has  no  relation  to  the  magnitude  of  the  surface  ; 

(6.)  Its  measure  is  much  less  than  in  the  case  of  sliding'sur- 
faces,  and  varies  in  the  inverse  ratio  of  the  diameter  of  the  rolling 
body. 

A  cylinder  of  lignumvitse  rolling  on  rulers  of  different  kinds 
of  wood,  and  having  a  diameter  of  32  inches,  was  not  resisted 
by  a  friction,  at  a  mean,  of  more  than  T£j . 

122.  The  friction  of  the  axles  of  wheels  is  still  of  another  des- 
cription ;  it  is  less  than  that  of  sliding,  and  more  than  that  of 
rolling  bodies.      It  follows,  in  all  respects,  the  general  laws  of 
sliding  bodies.      An  axle  of  iron,  turning  in  a  box  of  oak,  had  a 
friction  of  j  ;  when  both  were  of  wood,  the  friction  was  TV. 

In  metallic  axles,  resting  in  boxes  of  another  metal,  and  well 
coated  with  grease,  the  friction  has  been  found  to  be  no  more 
than  J, . 


Book  III.}  OF  FRICTION.  119 

Of  wooden  axles  in  wooden  boxes,  when  coated  in  a  similar 
manner,  from  T1^  to  ^T  . 

Of  iron  in  wood,  also  coated  in  grease,  oV- 

123.  To  adopt  the  theory  of  Coulomb,  friction  appears  to  arise 
from  the  porosity  of  bodies.  All,  even  the  most  dense,  have  large 
spaces  between  the  particles  of  which  they  are  composed.  When 
they  rest  upon  each  other,  the  prominent  parts  of  the  one  fall 
into  the  cavities  of  the  other,  and  are  in  a  manner  locked.  In 
hard  substances,  the  maximum  of  this  effect  will  be  produced  in 
a  short  time  ;  but  where  they  are  soft,  the  maximum  will  not  be 
reached  until  the  utmost  compression  the  pressure  is  capable  of 
producing,  is  attained. 

The  polishing  of  bodies  consists  merely  in  rubbing  down  the 
asperities,  and  multiplying  the  cavities  in  number,  but  diminish- 
ing their  depth  ;  hence  they  still  interlock ;  but  the  weight  must  be 
raised  but  a  small  distance,  in  order  to  disengage  the  prominences 
from  the  cavities,  and  the  more  perfect  the  polish  the  less  will  be 
the  height  to  which  the  weight  must  be  lifted.  The  friction  of  bo- 
dies in  motion  must  obviously  be  less  than  that  of  bodies  at  rest, 
because  the  projections  of  one  substance  require  a  certain  defi- 
nite time  to  adapt  themselves  to  the  cavities  of  the  other,  and  the 
difference  will  be  greatest  in  those  bodies  that  require  the  greatest 
time  to  adapt  themselves  to  each  other.  When  unctuous  matters 
are  interposed,  they  fill  up  the  cavities  and  prevent  the  penetra- 
tion of  the  prominent  parts  ;  hence  the  resistance  becomes  almost 
solely  that  which  is  due  to  the  attraction  of  their  own  particles. 
As  the  weight  increases,  liquids  being  more  readily  forced  out  of 
vessels  that  contain  them,  and  thus  most  easily  disengaged  from 
the  cavities,  oppose  less  resistance  than  solids  to  penetration,  and 
are  less  efficacious  in  diminishing  friction  ;  but  on  the  other  hand, 
the  more  perfect  the  fluidity,  the  more  easily  are  the  particles  of 
liquid  moved  among  each  other  ;  and  hence,  so  long  as  the  pres- 
sure is  not  sufficient  to  force  them  out  of  the  cavities,  they  will 
be  better  suited  to  diminish  friction  than  thicker  oils,  or  solid 
grease. 

The  particles  of  homogeneous  bodies  are  arranged  in  a  similar 
manner,  whether  by  the  action  of  crystallization,  or  their  organi- 
zation ;  hence  the  cavities  and  asperities  will  fit  better,  and  ap- 
ply themselves  more  closely,  than  in  heterogeneous  bodies. 

When  a  body  slides  upon  another,  one  of  two  things  must  occur, 
either  the  weight  must  be  partially  lifted,  or  the  asperities  must 
be  broken  down  before  the  sliding  body  ;  either  of  these  will  re- 
quire a  force  of  some  intensity,  but  when  a  body  rolls,  the  very 
act  of  rolling  disengages  the  prominences  from  the  cavities.  The 
action  of  axles,  resting  in  sockets,  is  intermediate  between  that  of 


120  OF    FBICTION.  [BOO/C  HI. 

rolling  and  that  of  sliding  bodies,  and  hence  has  an  intermediate 
degree  of  advantage.  It  will  be  seen  hereafter,  that  an  adhesion 
takes  place  between  bodies  which  depends  upon  the  extent  of  the 
surfaces,  and  although  this  be  extremely  small,  it  is  sufficient  to 
account  for  the  small  increase  that  follows  the  law  of  the  surface. 

The  friction  being  in  a  certain  degree  removed  from  the  bodies 
themselves,  and  taking  place  among  the  particles  of  the  oleaginous 
substances,  when  the  latter  are  used  as  coatings  ;  and  as  an  in- 
creased pressure  will  have  an  effect  in  overcoming  the  latter  re- 
sistance, it  appears  probable,  that  in  this  case,  an  increased  pres- 
sure may  act  in  opposition  to  the  other  part  of  the  friction,  of 
which  it  is  the  effective  cause. 

The  property  that  bodies  have  of  moving  forward  in  the  straight 
line  in  which  the  force  applied  has  been  directed,  would  prevent 
the  prominent  parts  of  bodies,  moving  with  greater  velocities, 
from  entering  as  deep  into  the  cavities  of  those  on  which  they 
move,'  as  they  would  if  moving  with  less  velocities,  and  hence,  at 
great  velocities,  there  ought  to  be  an  ascertainable  diminution  of 
the  friction. 

124.  The  whole  of  the  circumstances  that  are  involved  in  fric- 
tion, may  be  represented  by  .a  formula,  which  is  as  follows: 

F=/P— <pP4ys-yv.  (108) 

In  this,  F  represents  the  friction,  P  the  pressure,  S  the  area  of 
the  surface,  and  V  the  velocity  ;  f  is  the  ratio  of  the  pressure  to 
the  friction,  under  ordinary  circumstances,  say  the  fraction  given 
for  the  particular  cases  in  §120;  9  is  a  very  small  function  of  the 
pressure,  whose  amount  has  not  been  fully  established  from  ex- 
periment ;  <p'  is  the  cohesion  of  the  unit  of  the  surface  ;  and  <p"  a 
very  small  coefficient  of  the  velocity,  whose  magnitude  is  also 
unknown. 

125.  The  above  theory,  and  the  inferences  from  the  experi- 
ments whose  results  have  been  cited,   point  out  the  modes  that 
may  be  employed  in  practice  to  lessen  or  overcome  the  friction. 

(1.)  The  line  in  which  the  power  is  applied,  instead  of  being 
parallel  to  the  surface  on  which  the  body  moves,  may  be  slightly 
inclined  upwards.  If  the  power  be  resolved  into  two  components, 
one  of  which  is  parallel,  and  the  other  perpendicular  to  the  sur- 
face, the  first  alone  will  be  applied  to  the  draught,  and  the  other 
will  act  to  raise  the  weight  of  the  body.  By  this  latter  action, 
the  parts  of  the  surfaces  that  have  been  interlocked  may  be  dis- 
engaged. 

(2.)  The  highest  practicable  degree  of  polish  must  be  given  ; 
and  the  surfaces  except  in  the  case  of  rolling,  coated  with  unctuous 
matters.  When  the  pressure  is  small,  the  most  limpid  oils  should 
be  used  ;  as  the  pressure  increases,  those  of  more  viscidity  ;  on  a 


Book  ///.]  OP    FRICTIOBT.  121 

still  further  increase,  tallow  must  be^em ployed  ;  when  the  pres- 
sure becomes  very  great,  or  the  velocity  is  such  as  to  melt  the 
tallow,  plumbago  may  be  wfixed  with  that  substance  ;  and  finally, 
in  the  greatest  pressures  that  are  to  be  found  in  practical  mecha- 
nics, plumbago  alone,  or  soapstone,  both  in  the  state  of  fine  pow- 
der, may  be  employed. 

(3.)  The  motion  of  rolling  may  be  substituted  for  that  of  sli- 
ding, when  the  body  has  a  figure  that  will  admit  of  it. 

Thus  tobacco  in  Virginia  was  formerly  drawn  to  market,  by 
making  the  hogshead  roll  along  the  ground.  This  change  of  the 
mode  of  motion,  is  not  only  advantageous  in  bodies  of  a  circular 
section,  but  may  be  made  useful  in  others,  although  in  them  it 
will  become  necessary  to  lift  the  weight  at  each  turn  the  body 
makes.  If  the  force  necessary  to  lift  the  weight,  in  such  cases,  be 
not  greater  than  the  difference  between  rolling  and  sliding  fric- 
tions, an  advantage  will  be  gained. 

(4.)  If  the  body  be  of  such  a  figure  that  it  cannot  advantage- 
ously be  made  to  roll,  it  may  be  set  upon 'rollers,  and  the  ad- 
vantage which  is  due  to  their  diameters  and  mode  of  motion, 
will  be  attained.  The  body,  in  moving  forward,  will  leave  the 
rollers  behind  it,  which  must  therefore  be  lifted  and  carried  for- 
ward to  receive  it  again  ;  hence  this  method  is,  generally  speak- 
ing, confined  to  short  distances.  When,  however,  the  weight  is 
very  great,  it  may  be  more  advantageous  than  any  other  practica- 
ble means  :  thus,  for  instance,  the  great  rock,  that  forms  the  pe- 
destal of  the  statue  of  Peter  the  Great,  at  St.  Petersburgh,  was 
moved  a  distance  of  several  versts,  from  the  place  where  it  was 
found  imbedded,  to  the  banks  of  the  Neva,  upon  small  spheres  of 
metal,  laid  in  a  trough  ;  and  after  it  was  transported  to  the  city, 
was  carried  by  the  same  method  to  the  place  where  it  was  to  be 
set  up.  The  removal  of  this  vast  mass  would  have  been  imprac- 
ticable, by  any  of  the  ordinary  means  of  transportation. 

(5.)  When  rollers  are  inapplicable,  the  weight  to  be  moved 
may  be  laid  upon  a  wheel  carriage  :  here,  besides  the  diminution 
of  friction  which  is  obtained,  a  mechanical  advantage  is  gained, 
equivalent  to  the  ratio  of  the  diameter  of  the  wheel  to  that  of  its 
axle. 

(8.)  The  use  of  wheels  being  attended  with  this  mechanical 
advantage,  the  extremities  of  their  axles  may  be  made  to  rest 
upon  the  circumferences  of  other  wheels,  by  means  of  which  a 
similar  advantage  may  be  gained,  or  may  be  made  to  press  against 
revolving  bodies,  instead  of  resting  in  cylindrical  sockets.  Such 
applications  of  this  principle  are  called  friction  wheels,  and  fric- 
tion rollers. 


16 


122  o*  FRICTIOW.  [Book  III. 

The  most  beautiful  application  of  this  kind,  is  that  which  was 
adapted  by  Atvvood  to  his  machine,  which  has  already  been  spo- 
ken of  in  §  95.  In  this  apparatus,  the  extremities  of  the  axles 
of  the  wheel,  over  which  the  cord,  that  connects  the  weights, 
passes,  are  made  each  to  rest  upon  the  circumference  of  two 
other  wheels,  and  a  mechanical  advantage  is  gained  in  the  ratio 
that  has  just  been  mentioned.  By  this  arrangement,  the  friction 
is  rendered  wholly  insensible,  and  interferes  in  no  appreciable 
degree  with  the  results  of  the  experiment. 

In  the  patent  blocks  of  Garnett,  the  axles,  instead  of  moving 
in  cylindrical  sockets,  rest  each  upon  six  friction  rollers,  arranged 
in  a  box,  and  by  this  means  a  similar  advantage  is  gained. 

An  attempt,  founded  upon  similar  principles,  is  now  making 
by  Wynans  of  New-Jersey,  and  applied  to  the  wheels  of  car- 
riages. 

It  will  be  obvious,  that  the  advantage  of  wheels  ceases,  when 
their  effective  friction  becomes  greater  than  that  of  their  circum- 
ferences upon  the  surfaces  upon  which  they  move.  This  is  the 
case  upon  hard  smooth  substances,  such  as  ice,  in  which  case 
sledges  are  to  be  preferred  to  wheel  carriages. 

(7.)  A  knowledge  of  the  fact,  that  a  very  great  diminution  of 
the  rubbing  surface  is  attended  with  a  sensible  diminution  of  the 
friction,  may  be  applied  with  advantage  in  a  limited  number  of 
cases.  Thus  :  where  the  surfaces  are  so  hard  as  to  admit  of  no 
penetration,  even  where  progressive  motions  are  employed,  the 
moving  body  may  have  its  surface  diminished  almost  to  an  edge. 
Of  an  application  of  this  sort,  we  have  an  instance  in  the  case  of 
skates.  In  rotary  motions,  the  application  of  this  principle  is 
more  easy.  Thus  :  although  in  machinery,  generally,  the  axles 
must  be  of  a  certain  size,  in  order  to  bear  the  weight  of  the 
wheels,  and  often  the  action  of  other  pressures  to  which  they  are 
subjected,  these  weights  and  pressures  become  so  small  in  the 
case  of  part  of  the  train  of  wheels  in  the  common  watch,  that 
the  bearing  of  the  axles  may  be  reduced  to  the  smallest  points 
that  can  be  made  on  hardened  steel.  These  points,  instead  of 
resting  in  sockets,  are  supported  in  small  shallow  cups  of  a  hard 
material,  agate,  ruby,  or  diamond.  In  motions  of  oscillation  upon 
axes,  the  axis  may  take  the  form  of  an  edge  of  steel,  and  may  be 
made  to  rest  upon  a  cylindrical  surface,  or  even  upon  a  polished 
plane  of  some  hard  material ;  of  this  we  shall  have  instances  in 
the  Balance  and  the  Pendulum. 

126.  Friction  is  by  far  the  most  influential  of  the  causes,  by 
which  bodies  moving  near  the  surface  of  the  earth  are  brought 
to  rest.  If  supported,  they  experience  a  friction  from  the  body 
that  supports  them  ;  if  unsupported,  they  fall  to  the  earth,  either 


///.]  OP    FRICTION.  123 

in  a  vertical  or  inclined  direction ;  if  in  a  vertical  direction,  the 
friction  they  meet  in  penetrating,  rapidly  destroys  their  motion, 
even  if  the  earth  be  soft  where  they  fall :  if  in  an  inclined  direc- 
tion, of  the  two  components  of  their  metion,  one  of  which  is 
perpendicular,  the  other  parallel  to  the  surface  of  the  earth,  the 
former  is  at  once  destroyed  by  the  resistance  to  penetration,  the 
other  remains  to  carry  the  body  along  the  surface,  and  this  again 
would  be  finally  destroyed,  by  friction  against  the  surface,  even 
did  no  other  retarding  force  act. 


124  OT    THE    STIFFNESS  [Book  III. 

CHAPTER  T. 
OF  THE  STIFFNESS  OF  ROPES. 

127.  It  frequently  becomes  necessary,  in  practical  mechanics, 
to  bend  ropes  over  cylinders  and  rollers.     This  is  the  case  even 
in  some  of  the  elementary  machines.     When  ropes  are  thus  bent, 
they  always  oppose  a  resistance,  to  overcome  which  it  becomes 
necessary  to  apply  a  part  of  the  force  that  actuates  the  machine. 
This  resistance,  it  is  obvious,  may  vary  : 

(1.)  With  the  tension  of  the  rope,  or  the  weight  by  which  it 
is  stretched ; 

(2.)  With  the  quality  of  the  rope,  depending  upon  the  nature 
of  its  materials,  and  the  manner  of  its  manufacture  5 

(3.)  With  the  size  of  the  rope  ; 

(4.)  With  the  diameter  of  the  cylinder,  over  which  it  is  bent. 

128.  The  best  experiments  on  this  subject  are  also  by  Cou- 
lomb.    The  first  of  the  results  obtained  by  him  is  :  that  like  fric- 
tion, the  resistance  of  ropes  to  forces  applied  to  bend  them,  is  a 
constant  retarding  force.     The  deductions,  in  respect  to  the  cir- 
cumstances that  have  been  stated,  are  as  follows,  viz.  : 

(1.)  The  resistances  of  ropes  are  directly  as  the  tensions  to  which 
they  are  subjected  ; 

(2.)  The  resistance  is  greatest  in  ropes  that  have  been  strongly 
twisted,  in  ropes  coated  with  tar,  and  in  new  ropes.  The  ratio  of 
these  is  in  some  measure  included  in  the  next  circumstance. 

(3.)  The  resistance  increases  with  some  determinate  power  of 
the  diameter  of  the  rope,  which  we  shall  call  n  : 
In  new  tarred  ropes,  n=2, 
In  new  white  ropes,    n=1.7, 
In  old  ropes,  n=1.5. 

(4.)  The  resistances  are  inversely  as  the  diameters  of  the  cylin- 
ders, around  which  the  ropes  are  bent. 

129.  When  a  rope  is  wound  more  than  once  around  a  cylinder, 
it  is  found  that  the  resistances  increase  in  geometric  progression. 

This  principle  is  frequently  applied  in  practice,  when  it  is 
wished  to  oppose  rapidly  increasing  resistances  to  moving  bodies  : 
thus,  in  arresting  the  progress  of  a  vessel,  a  rope  is  turned  again 
and  again  around  a  post ;  and  a  small  number  of  turns  will  be- 
come efficient  to  overcome  any  force  not  of  sufficient  intensity 
to  break  the  rope. 

The  resistance,  in  any  particular  case,  is  represented  by  the 
formula, 


Book  III.}  o»  RO?US.  125 

rn+pw,  (109) 

when  m  is  the  absolute  resistance  of  the  rope,  and  p  the  propor- 
tion of  the  weight  w,  that  is  necessary  to  be  added,in  order  to  over- 
come the  increased  resistance  due  to  the  addition  of  a  weight. 

1  30.  To  furnish  data  for  the  application,  itis  sufficient  to  quote  a 
single  instance  calculated  from  the  experiments  of  Coulomb, 
whence,  by  the  principles  we  have  laid  cown,  the  resistance  of 
any  other  rope  may  be  calculated.  A  roje  not  tarred,  of  an  inch 
in  diameter,  may  be  bent  around  a  cylinder  of  4  inches  in  diame- 
ter, by  a  weight  of  TVth  of  a  pound,  adced  to  TW  of  the  weight 
b)'  which  it  is  strained  ;  a  tarred  rope,  o:%  the  same  size,  requires 
a  weight  of  about  ^th  of  a  pound,  added  .o  J-yth  of  the  weight  by 
which  it  is  strained.  From  these  two  iistances,  the  resistance  of 
any  other  rope  may  be  calculated  by  mtans  of  the  principles  that 
have  been  laid  down. 

The  sizes  of  ropes  are  usually  estinated  by  the  measures  of 
their  circumference,  as  1,  2,  3  inch,  &c.  Hence  the  instance* 
given  may  serve  as  units. 

The  resistance,  in  all  cases  whatsoevtr,  will  be 


, 

In  which  expression,/?,  w,  and  m  are  35  in  the  former  equation, 
and  have  the  values  given  in  our  instance  ;  n  is  the  power  from 
§128  ;  cthe  circumference  of  the  rope;  ind  d  the  diameter  of  the 
.cylinder,  both  expressed  in  inches, 


126  OP  THE  [Book  III. 


CHAPTER  VI. 

E  MECHANIC  POWERS. 

131.  A  machine  isaninstrument,by  means  of  which  we  change 
either  the  direction  onthe  intensity  of  a  force,  or  both  its  di- 
rection and  intensity.  The  general  principle  of  the  equilibrium 
of  all  machines  whatsoever,  is  to  be  found  in  that  of  virtual  veloci- 
ties, By  this,  if  the  sevral  points  of  the  machine,  on  which  the 
forces  act,  were  each  to  \e  supposed  to  move  under  the  action  of 


the  force  that  is  applied, 
the  products  of  all  the  i 
latter  being  distinguishe 
their  directions,  is  equal  I 
have  a  fixed  point,  the  pr 
5th,  becomes :  Equilibri 
sum  of  the  products  of  all 
into  the  respective  virtua 


jquilibrium  will  exist  when  the  sum  of 
rces  into  their  several  velocities,  the 
as  positive,  or  negative,  according  to 
0.     As  machines,  generally  speaking, 
position,  in  conformity  with  §  71,  case 
m  will  exist  in  any  machine,  when  the 
c  forces,  on  each  side  of  the  fixed  point, 
velocities  of  their  points  of  application. 


is  exactly  equal  to  the  sum  of  the  similar  products  on  the  oppo- 
site side  of  the  fixed  poir,t.  The  value  of  this  principle,  as  ap- 
plied to  the  useful  propeiiies  of  machines,  will  be  discussed  here- 
after. For  the  present,  Ive  shall  leave  it,  and  proceed  by  more 
direct  methods  to  investigate  the  conditions  of  equilibrium  in 
the  more  simple  forms  of  machines. 

132.  Machines  are  either  simple  or  compound.     The  former 
are  the  elementary  parts,  of  which  all  compound  machines  are 
made  up,  by  combinations  of  various  descriptions  ;  they  are  also 
capable  of  being  used  singly.     These  simple  machines  are  called 
the  Mechanic  Powers. 

133.  The  Mechanic  Powers  are  six  in  number,  viz.  :  The  Le- 
ver, the  Wheel  and  Axle,  the  Pulley,  the  Wedge,  the  Inclined 
Plane,  and  the  Screw. 

It  has  already  been  stated,  that  their  properties  may  be  re- 
duced to  a  single  principle  ;  but  in  the  mode  that  has  been  chosen 
for  examining  their  conditions  of  equilibrium,  it  will  be  seen, 
that  the  properties  of  four  of  them  may  be  included  in  those  of 
two  others.  Hence  the  Mechanic  Powers  are  arranged  in  two 
divisions :  to  the  first  belong  the  Lever,  the  Wheel  and  Axle, 
and  the  Pulley ;  to  the  second,  the  Wedge,  the  Inclined  Plane, 
and  the  Screw. 


MECHANIC    POWERS.  127 

Oftfe  Lever. 

134.  A  lever  is  an  inflexible  bar,  or  rod,  resting  upon  a  fixed 
axis  or  prop,  that  is  called  the  Fulcrum,  around  which  it  is  free 
to  move,  under  the  action  of  the  impressed  forces.     The  general 
condition  of  equilibrium,  in  the  case  of  any  number  of  forces 
whatsoever,  is  an   immediate  deduction  from  the  theory  of  the 
moments  of  rotation  in  §  34,  and  is  as  follows  : 

135.  In  any  lever,  whatsoever,  equilibrium  will  exist,  when 
the  sum  of  the  products  of  the  forces  applied  to  it,  on  one  side  of 
the  fulcrum,  into  the  perpendiculars  let  fall  from  that  poi-nt  upon 
their  respective  directions,  is  exactly  equal  to  the  sum  of  similar 
products,  on  the  opposite  side  of  the  fulcrum. 

Among  the  forces,  it  is  obvious,  that  the  weight  of  the  lever 
itself  may  be  included,  its  direction  being  a  vertical  line,  and  its 
point  of  application  the  centre  of  gravity  of  the  bar. 

136.  The  properties  of  the  lever,  and  indeed  of  all  the  simple  me- 
chanic powers,  are  most  usually  limited  to  the  case  of  the  action 
of  no  more  than  two  forces.   One  of  these  is  called  the  Power,  the 
other,  the  Weight.  By  the  Weight,  we  understand  the  resistance 
to  he  overcome  ;  by  the  Power,  the  force,  whatever  be  its  nature, 
that  is  applied  to  overcome  the  resistance.     In  the  usual  mode  of 
treating  the  theory,  the  bar  is  supposed  to  be  devoid  of  weight. 
The  points  of  application  of  these  two  forces,  and  the  fixed  point, 
or  fulcrum,  may  have  three  possible  positions  in  respect  to  each 
other,  and  we  hence  distinguish  three  different  kinds  of  lever. 

(1.)  WThen  the  fulcrum  is  between  the  power  and  the  weight. 
(2.)  When  the  weight  is  between  the  power  and  the  fulcrum. 
(3.)  When  the  power  is  between  the  weight  and  the  fulcrum. 

137.  If  the  lever  be  straight,  and  the  power  and  weight  act 
parallel  to  each  other,  equilibrium  will  exist,  when  the  power 
is  to  the  weight  in  the  inverse  ratio  of  their  respective  distances 
from  the  fulcrum; 

or  when  Pa=W6.  (Ill) 

This  case  becomes  the  simple  one  in  §  22,  of  finding  the  point 
of  application  of  the  resultant  of  two  parallel  forces ;  for  the  fixed 
point  will  be  in  the  same  state,  as  if  it  were  acted  upon  by  a  force 
equal  and  opposite  to  the  resultant  of  the  other  two  ;  and  it  was 
there  demonstrated,  that  the  point  of  application  of  the  resultant 
of  two  parallel  forces,  divides  the  line  of  application  into  parts 
inversely  proportioned  to  the  intensities  of  the  two  forces. 

138.  In  the  case  of  a  bent  lever,  or  when  upon  a  lever,  whether 
straight  or  crooked,  the  directions  of  the  power  and  weight  are 
not  parallel,  equilibrium  will  exist  when  the  two  forces  are  to 


128  OP  THE  [Book  IIL 

each  other  inversely  as  the  perpendiculars  let  fall  from  the  ful- 
crum upon  their  respective  directions. 

This  may  be  deduced  directly  from  (25),it,beinfi;  again  obvious, 
as  in  the  former  case,  that  the  fulcrum  must  be  the  point  of  the 
application  of  the  resultant  of  the  two  forces  that  keep  the  lever 
in  equilibrio ;  and  by  that  equation  it  is  shown,  that  this  point  is 
situated  in  such  a  manner,  that  the  perpendiculars  let  fall  from  it, 
upon  the  directions  of  the  two  forces,  are  in  the  inverse  ratio  of 
the  intensities  of  the  two  forces. 

139.  The  lever  is  perhaps  the  most  ancient,  as  it  is  still  the 
most  familiar  in  its  use  of  all  the  mechanic  powers.  The  instances 
of  its  practical  application  are  almost  too  numerous  to  admit  of 
their  being  named.  It  will  be  sufficient  to  give  a  few,  merely  as 
illustrations  of  its  properties. 

(1.)  Of  the  first  kind  of  lever,  where,  as  in  the  figure  beneath, 
the  fulcrum  is  between  the  power  and  the  weight,  we  have  in- 


stances in  the  common  crow-bar  and  handspike,  in  scissors,  po- 
kers, pincers,  snuffers;  and  of  a  bent  lever  of  this  description,  in 
a  hammer,  when  used  for  drawing  nails. 

(2.)  In  oars,  the  fulcrum  is  where  the  blade  strikes  the  water  ; 
the  weight  is  applied  where  the  oar  rests  against  the  side,  and  the 
power  is  applied  by  taking  hold  of  the  opposite  extremity  ;  hence 


they  are  levers  of  the  second  kind,  in  which,  as  in  the  figure,  the 
power  is  between  the  fulcrum  and  the  weight. 

The  rudders  of  ships  act  upon  similar  principles.  Of  the  same 
kind  of  lever,  cutting  knives  fixed  at  one  end,  doors  moving  upon 
their  hinges,  the  manner  in  which  a  weight  is  borne  upon  a  wheel- 
barrow, nut-crackers,  &c.,  may  be  cited  as  instances.  In  these 
two  first  kinds  of  lever,  the  distance  of  the  power  from  the  ful- 
crum is  greater  than  that  of  the  weight;  hence  the  weight  has  a 
greater  intensity  than  the  power,  when  the  two  are  in  equilibrio, 
and  thus  the  power  is  capable  of  overcoming  a  resistance  greater 
than  its  own  measure. 


Book  ///.]  MECHANIC  POWERS.  129 

(3.)  Of  the  third  kind  of  lever,  in  which,  as  represented  on 
the  figure,  the  power  is  between  the  weight  and  the  fulcrum,  we 


have  examples  in  the  common  tongs,  in  sheep  shears,  in  the  man- 
ner a  ladder  is  raised  against  a  wall.  In  this  kind  of  lever,  the 
power  being  nearer  to  the  fulcrum  than  the  weight,  the  former 
must  have,  in  the  case  of  equilibrium,  a  greater  intensity  than 
the  latter.  In  the  two  first  kinds  of  lever,  then,  a  given  power 
will  raise  a  greater  weight,  or  overcome  a  greater  resistance  than 
it  can  when  it  acts  directly;  while,  in  the  third  kind  of  lever, 
the  weight  raised,  or  the  resistance  overcome,  is  less  than  the 
power  is  capable  of  doing,  if  it  act  without  the  intervention  of 
the  lever.  If  the  several  figures  be  referred  to,  it  will  be  seen 
that  these  advantages  in  the  two  first  cases,  and  this  disadvantage 
in  the  last,  are  compensated  by  the  difference  in  the  velocities  of 
the  points  of  application,  in  case  motion  should  take  place.  These 
velocities  will  be  represented  by  the  arcs  described.  These  are  to 
each  other  as  the  radii  of  the  respective  circles,  of  which  they 
are  similar  arcs,  and  as  their  radii  are  the  arms  of  the  levers,  that 
are  to  each  other  in  the  inverse  ratio  of  the  forces  applied  at  their 
extremities,  the  product  of  the  weight  into  its  velocity,  will  be 
equal  to  that  of  the  power  into  its  velocity.  This,  it  will  be  at 
once  seen,  is  no  more  than  a  case  of  the  general  principle  of  vir- 
tual velocities.  Hence,  whenever  intensity  of  force  is  gained  by 
means  of  the  lever,  it  is  always  at  the  expense  of  an  equal  loss  of 
velocity;  and  where  velocity  is  gained,  it  is  gained  at  the  ex- 
pense of  power.  A  similar  inquiry  into  the  velocities,  with  which 
the  points  of  application  of  the  power  and  weight  would  move,  in 
the  case  of  the  equilibrium  being  disturbed,  would  show  that 
the  same  relation  exists  between  the  intensity  offeree,  and  the 
velocities  lost  or  gained  in  actual  motions. 

140.  The  Balance  is  one  of  the  most  useful  applications  of  the 
lever.  It  is  no  more  than  a  lever  of  the  first  kind,  with  arms  of 
equal  lengths,  resting  upon  a  fulcrum.  To  the  two  extremities 
are  attached  pans,  or  scales,  in  which  heavy  bodies  may  be  placed. 
It  will  be  obvious,  that  when  the  two  weights  are  equal,  the  ba- 
lance will  be  in  equilibrio  under  their  joint  action.  If  then  a 
certain  number  of  units  and  fractions  of  any  conventional  system 

17 


13*  0*  TSX  [BOO*  W 

of  standard  weights,  be  in  one  of  the  scales,  it  will  just  counter- 
poise a  substance  placed  in  the  other,  whose  weight  must  have  an 
equal  value  in  that  system.  Hence,  by  means  of  a  balance,  the 
unknown  weight  of  any  articles  whatever,  may  be  determined 
by  the  aid  of  a  set  of  properly  graduated  weights. 

The  balance,  having  this  property,  is  of  the  most  extensive 
utility,  not  only  in  philosophical  inquiries,  but  also  in  every  va- 
riety of  trade,  in  which  the  articles  cannot  have  their  quantities 
determined  by  measures  of  length,  of  surface,  or  of  capacity, 
either  on  account  of  their  absolute  nature,  or  in  compliance  with 
custom. 

141.  It  is  difficult,  in  practice,  to  make  the  distances  between 
the  two  points  whence  the  scales  are  suspended,  and  the  fulcrum, 
exactly  equal.  Still  a  sufficient  degree  of  accuracy  may  be  at- 
tained for  all  the  purposes  of  trade.  But  it  sometimes  happens 
that  balances,  either  by  accident  or  design,  have  arms  of  unequal 
lengths.  In  this  case,  the  error  may  be  detected  by  changing 
the  position  of  the  two  counterpoising  weights  from  one  scale  to 
the  other.  If,  when  thus  transferred,  they  are  still  in  equilibrio, 
the  balance  is  true,  if  they  are  not,  it  is  false. 

The  actual  weight  may  be  obtained  by  means  of  a  false  balance, 
by  weighing  the  substance  whose  quantity  is  to  be  determined, 
successively,  in  the  two  scales.  Its  true  weight  will  be  the  geo- 
metric mean  between  the  known  weights  that  counterpoise  it,  in 
the  two  different  positions. 

of  the  nfcotoce  vim  m**r 


scales;  •  and  o  the  tiro  aims :  then 
nie  property  of  uw  lever, 

P«=W4,  (111) 


and  W  :  P  :  W.  (112} 

When  a  balance  is  used  for  delicate  investigations,  any  error 
that  might  arise  from  an  inequality  in  the  arms,  is  readily  ob- 
viated by  a  simple  process.  The  body,  whose  weight  is  to  be 
determined,  b  placed  in  one  of  the  scales,  and  is  counterpoised 
by  a  substance  capable  of  minute  division,  such  as  fine  sand,  placed 
in  the  other.  When  the  balance  has  been  brought  to  rest  in  a 
truly  horizontal  position,  the  body  to  be  weighed  Is  removed,  and 
weights  placed  in  the  same  scale,  until  they  counterpoise  the 
substance  that  remains  in  the  other  scale.  It  will  be  obvious, 
that  the  body  whose  weight  is  required,  and  the  standard  weights 


Book  in.]  XXCKAXK  rOWXML  131 

being  both  in  equilibrio  with  the  same  pihiiiiirr,  and  having 
acted  upon  the  same  arm  of  the  lever,  must  he  exactly  equal  to 
each  other. 

142.  The  properties  of  a  good  balance  are — 

(1.)  That  it  should  rest  io  a  horizontal  position  when  Inajij 
with  equal  weights,  and  in  an  '"fljpf^  pfreitti?"  when  the  weights 
are  not  equal ; 

(2.)  That  it  should  have  great  sensibility,  so  that  a  small  pro- 
portion of  the  weight  with  which  it  is  loaded,  added  to  either 
scale,  shall  disturb  the  equilibrium  ; 

(3.)  That  it  should  be  stable,  or  soon  return  to  rest,  after  bar- 
ing been  put  in  motion  by  a  change  of  the  weijfes. 

These  properties  depend  in  part  upon  a  proper  choice  of  the 
point  of  suspension,  in  respect  to  the  positions  of  the  line  that 
joins  the  points  whence  the  scales  are  suspended,  and  of  the  cen- 
tre of  gravity ;  and  in  part  upon  accurate  mechanical  construc- 
tion. 

In  the  figure  beneath,  let  A  and  B  be  the  points  whence  the 


scales  are  suspended;  G,  the  centre  of  gravity;  O,  the  centre  of 
suspension ;  C,  the  point  where  the  lines  that  join  AJ3,  and  O,G, 
intersect  each  other. 

If  the  points  O,C,G,  were  to  correspond,  the  balance  would 
be,  §  1O7,  in  a  state  of  indifference,  it  would  be  the  most  JIHIJ 
ble  to  variations  of  weight,  but  would  have  no  tendency  to  come 
to  rest  in  a  horizontal  position.  The  higher  the  point  O,  the 
more  stable  will  be  the  equilibrium,  but.  the  less,  all  things  else 
being  equal,  will  be  the  sensibility.  The  longer  the  arms,  the 
greater  will  be  the  moments  of  rotation  of  the  weights,  and  con- 
sequently the  greater  the  sensibility.  The  centre  of  gravity  will 
be  lowered  by  additional  weights  in  the  scales,  and  in  this  way 
also  the  sensibility  will  be  diminished. 

If  the  point  O  should  Call  below  C,  the  balance  will  be  unsteady; 


133  OF  THE  [Book  HI. 

but  if  below  both  C  and  G,  the  balance  will  be  in  a  state  of  tot- 
tering equilibrium,  and  will  be  liable  to  be  overturned. 

In  order  that  the  balance  shall  come  to  rest  in  a  horizontal  po- 
sition, not  only  must  its  arms  be  of  equal  lengths,  but  they,  with 
the  scales  attached,  must  be  of  equal  weights. 

In  the  actual  construction  of  the  balance  it  is  important — 

(1.)  That  the  motion  shall  be  attended  with  as  little  friction  as 
possible  :  this  is  effected  by  making  the  axis  of  suspension  of  the 
form  that  is  called  a  knife-edge.  A  prismatic  bar  of  hard  steel  is 
passed  through  the  beam  of  the  balance,  and  is  formed  into  an 
edge  beneath,  by  the  intersection  of  two  convex  curves.  In  the 
common  balance,  this  is  made  to  rest  at  each  end  on  the  surface 
of  a  hollow  ringj^r  cylinder. 

In  some  of  the  more  accurate  balances,  the  knife-edges  rest 
upon  planes  of  polished  agate.  This  is  one  of  the  cases  that  were 
quoted  in  §  125,  in  which  the  friction  is  reduced  to  the  minimum, 
by  diminishing  the  rubbing  surface  to  a  mere  edge. 

As  the  friction  is  in  the  balance,  as  in  other  practical  cases,  pro- 
portioned to  the  pressure,  the  greater  the  weight  with  which  the  ba- 
lance is  loaded,  the  less  will  be  the  sensibility  ;  and  the  latter  is,  as 
has  just  been  shown,  also  diminished,  by  the  lowering  of  the  centre 
of  gravity.  To  obviate  this  defect,  some  balances  have  been  made 
with  a  sliding  weight  beneath  the  point  of  suspension,  by  chang- 
ing the  position  of  which,  the  balance  may  be  made  more  or  less 
sensible. 

(2.)  The  distance  hp.tween  the  centre  of  suspension,  and  the 
points  whence  the  scales  hang,  ought  10  remain  exactly  the  same 
during  all  the  oscillations  of  the  balance.  This  is  sometimes  ef- 
fected by  suspending  the  scales  from  knife-edges  also.  These  are 
passed  through  the  ends  of  the  beam  with  the  edges  uppermost  ; 
and  the  scales  are  hung  from  hooks  or  rings  that  rest  upon  them. 
Sometimes,  to  make  the  touching  surfaces  the  least  possible,  these 
rings  are  ground  on  the  inside  to  a  sharp  edge. 


Book  III.]  MECHANIC  POWERS.  13S 

A  balance  of  a  good  construction  is  represented  beneath. 


In  some  of  the  best  balances  by  Ramsden  and  Troughton,  the 
beam,  instead  of  being  a  bar,  is  made  of  the  form  of  two  similar 
and  equal  hollow  cones,  joined  together  at  their  greater  bases. 


Such  a  figure  possesses  far  more  strength  than  a  solid  bar.  It 
may,  therefore,  be  made  much  lighter  than  almost  any  other 
form.  It  is  said  that  these  instruments  weigh  so  well,  as  to  not$ 


13*  op  THE  {Book  III 

differences  of  one-millionth  part  of  the  weight  with  which  the 
scales  are  loaded. 

It  is,  however,  an  excellent  balance  that  will  weigh  to  the 
TffTFoinrth  of  the  weight  with  which  it  is  loaded,  and  generally 
speaking,  balances  do  not  weigh  more  nearly  than  from  ^j^  to 
smooth  of  the  weight. 

Balances  of  different  sizes,  strength,  and  materials,  have  differ- 
ent degrees  of  accuracy.  For  weighing  weights  of  different  mag- 
nitudes, then,  several  balances  will  be  necessary,  from  those  which 
turn  with  weights  of  a  small  fraction  of  a  grain,  to  those  which 
will  bear  several  tons. 

143.  Levers  may  be  combined  together  in  such  a  manner  as  to 


forma  compound  machine,  as  is  the  case  in  the  figure. 

Here  the  power  P  would  be  in  equilibrio  with  a  force  acting 
at  W,  when  their  relation  was  inversely  as  their  distances,  or 


but,  if  instead  of  a  weight,  the  end  W  be  made  to  act  upon  the  ex- 
tremity of  another  lever,  whose  arms  are  p'  and  w\  then 

Wj/=  WV  ; 

and  the  third  lever  will  have  the  following  condition  of  equili- 
brium : 


whence 

Ppp'p"=Www'w":  (113) 

therefore, 

In  a  combination  of  levers,  the  power  will  be  in  equilibrio 
with  the  weight,  when  the  former  is  to  the  latter,  as  the  con- 
tinued product  of  all  the  arms  of  the  lever,  on  which  the  weights 
act,  is  to  the  continued  product  of  all  the  arms  on  which  the 
power  acts. 

144.  Combinations  of  levers  may,  upon  this  principle,  be  used 
as  weighing  machines:  for  if  the  relation  between  the  lengths  of 
their  several  arms  be  known,  the  relation  between  a  known 
weight,  acting  at  one  extremity  of  the  first  lever,  to  the  unknown 
mass  which  is  in  equilibrio  with  it  at  the  farthest  point  of  the 
system,  is  also  known  ;  and  the  weight  of  the  latter  will  be  deter- 
minable.  The  most  useful  application  of  such  a  combination  of 
levers  is  in  the  platform  balance,  of  which  the  following  is  a  des- 
cription : 


Book  III.] 


MECHANIC    POWERS. 


135 


AB  is  a  section  of  a  platform  of  wood,  of  a  rectangular  shape, 
resting  upon  a  frame  represented  at  H  and  I.  CP",  C'P",  are  levers 
of  the  second  class,  having  their  fulcrums  at  C  and  CV  Of  these 
there  are  four,  diverging  from  the  centre  of  the  platform,  in  the 
direction  of  i-ts  semi-diagonals,  to  the  four  corners.  At  W  and 
\V,  upon  the  two  that  are  represented  in  the  section,  are  pins, 
which,  by  a  small  motion,  may  be  brought  in  contact  with  the 
platform,  and  thus  may  be  made  to  raise  it  from  the  frame,  and 
bear  its  weight  with  that  of  the  articles  to  be  weighed.  The  ex- 
tremities of  these  four  levers  rest  upon  a  bar  at  P",  which  is  sup- 
ported at  the  point  W",  by  another  lever  of  the  second  class  DP'. 
The  last  is  connected  by  a  wire  reaching  from  the  extremity 
P',  of  its  longer  arm,  to  the  shorter  arm  of  a  lever  of  the  first  class  ; 
at  the  opposite  end  of  which  a  scale,  S,  is  suspended.  A  weight 
placed  in  the  scale  S,  will  raise  the  point  P',  and  with  it  the  bar 
P";  and  the  rise  of  the  latter  will  cause  the  weight  of  the  plat- 
form, and  the  articles  with  which  it  is  loaded,  to  press  upon  the 
pins  W,  W.  Thus  a  small  weight  in  the  scale,  S,  will  be  in 
equilibrio  with  a  large  one  on  the  platform  AB,  and  their  rela- 
tion will  be  given  by  the  formula  (113).  The  lever,  P'D,  is 
generally  placed  in  a  position  at  right  angles  to  that  in  which  it 
is  represented,  projecting  beyond  one  of  the  longer  sides  of  the 
rectangle  :  and  it  is  evident,  from  mere  inspection,  that  the  four 
levers,  P"C,  &c.  act,  so  far  as  the  conditions  of  equilibrium  are 
concerned,  as  a  single  one.  Four  are  used,  in  order  to  bear  the 
platform  at  a  sufficient  number  of  points,  and  cause  it  to  rise  and 
fall  parallel  to  itself. 

145.  A  Steelyard  is  another  modification  of  the  first  kind  of  le- 
ver, which  is  also  used  as  a  weighing  machine.  The  lever  in  this 
case  has,  as  represented  in  the  figure  on  the  next  page,  unequal  arms. 
To  the  shorter  of  these,  the  substance  to  be  weighed  is  attached,  and 


136  OJ    THE  [Book  III. 

its  weight  is  determined  by  means 
of  a  constant  known  weight,  that  is 
moved  to  different  distances  from 
the  fulcrum,  until  it  be  in  equili- 
brio  with  the  substance  to  be  weigh- 
ed. If  their  distances  from  the  ful- 
crum be  equal,  the  two  weights  are 
equal  ;  if  the  constant  weight  be  twice  as  far  from  the  ful- 
crum as  the  fixed  point,  whence  the  substance  to  be  weighed 
is  suspended,  it  will  be  obvious  that  the  latter  weighs  twice  as 
much  as  the  former  :  and  thus,  as  the  distance  of  the  constant 
weight  varies  in  arithmetical  progression,  the  unknown  weight 
wilf  vary  as  those  distances.  The  longer  arm  of  the  steelyard  is 
therefore  cut  into  equal  divisions,  and  the  unknown  weight 
is  determined  by  the  distance  at  which  the  constant  moveable 
weight  is  from  the  fulcrum,  at  the  time  equilibrium  takes  place. 

The  constant  weight  is  suspended  from  the  lever,  by  means  of 
a  hook  or  ring  that  is  cut  beneath  into  a  sharp  edge,  to  enable  it 
to  adapt  itself  to  the  notches  that  form  the  divisions  of  the  longer 
arm.  Hence  there  is  danger  of  these  divisions  being  cut  and 
widened,  until  the  instrument  ceases  to  give  true  indications.  So, 
also,  when  the  lever  is  inclined,  the  distance  of  the  constant  weight 
varies.  In  few  cases,  indeed,  can  the  steelyard  be  depended  upon 
for  giving  as  true  a  measure  of  weight  as  the  balance.  If  a  scale 
be  suspended  by  knife-edges,  from  the  longer  arm  of  an  unequal 
lever,  weights  placed  in  it  will  have  a  value  in  determining  the 
weight  of  a  substance  suspended  from  the  shorter  arm,  as  much 
greater  than  their  true  value,  as  the  shorter  arm  is  less  than  the 
longer.  Upon  this  principle  they  may  be  graduated  ;  and  a  weigh- 
ing machine,  thus  constructed,  will  have  an  advantage  over  a 
balance  when  very  great  weights  are  to  be  determined  ;  for  the 
short  arm  will  be  less  liable  to  break  under  their  action  than  the 
arm  of  a  balance,  and  the  load  upon  the  knife-edges  will  be  much 
less. 

146.  Were  there  no  resistance  to  the  motion  of  the  lever,  it 
could  be  set  in  motion  by  the  smallest  addition  either  to  the  pow- 
er or  the  weight;  but  in  this  case,  as  in  all  others,  friction  will 
interpose  ;  and  if  the  weight  be  set  in  motion  by  the  power,  the 
latter  must  first  receive  an  addition  to  its  intensity,  when  merely 
in  equilibrio,  which  is  equal  to  the  moment  of  rotation  of  the 
friction  ;  and  in  case  it  is  desired  that  the  weight  shall  set  the 
power  in  motion,  the  former  must  first  receive  a  similar  addition, 
in  order  to  be  ready  to  cause  motion  by  the  smallest  new  accession 
of  force. 


Book  ///.]  MECHANIC    POWERS'. 

In  the  lever,  the  general  equation  of  equilibrium  is 


137 


if  we  let  the  resistance,  whatever  be  its  nature,  =R,  and  let  Rr  be 
its  moment  of  rotation,  then  in  the  case  in  which  the  system  is 
ready  to  be  set  in  motion, 

Pp=Ww^Rr,     .  (114) 

which  will  be  a  general  condition  in  all  machines  whatsoever. 
To  apply  this  to  the  case  of  the  lever  : 


Let  AB  be  the  lever  acted  upon  by  the  parallel  forces  P  and  W, 
and  turning  upon  tjie  cylinder  C,  as  an  axle  ;  call  the  arms  p  and 
ic.  Let  the  power,  P,  be  on  the  point  of  setting  the  weight,  W,  in 
motion.  The  pressure  on  the  axle  is  equal  to  the  resultant  of 
the  two  forces,  P  and  W  ;  resolve  this  resultant  into  two  forces, 
one  of  which  is  a  tangent  to  the  axle,  the  other  a  normal.  Let 
m  be  the  angle  the  tangential  force  makes  with  the  direction  of 
the  resultant,  the  two  components  are 

(P+W)  cos.  m, 

(P+W)  sin.  m. 

that  part  of  the  friction  which  is  directly  opposed  to  the  motion, 
will  have  the  same  ratio  to  the  whole  friction,/,  as  the  latter  of  these 
components  to  the  whole  pressure,  or  will  be 
/(P+W)  sin  m, 

1 
and  sin.  m~  ^/  (\-l-f  *\  » 

whence  the  friction  becomes 

/(P+W) 


and  as  r  is  its  distance  from  the  centre  of  motion,  Rr  in  the  gene- 
ral formula  (114)  becomes 


.  . 

and  the  condition  of  the  state  of  the  machine,  in  which  motion  is 
about  to  begin,  is 


1S8  or  THE  [Book  III 

The  co-efficient,/,  is  in  most  cases,  so  small  a  fraction,  that  its 
square,/2,  may  be  neglected,  in  which  case  the  formula  becomes 

(118) 


Of  the  Whetl  and  Axle. 

147.  The  Wheel  and  Axle,  as  its  name  imports,  is  a  wheel 
firmly  connected  to  an  axle,  and  moving  with  it  upon  a  common 
axis.  The  power  is  applied  to  the  circumference  of  the  wheel  ; 
the  weight  to  the  circumference  of  the  axle.  It  may,  there- 
fore, be  considered  as  a  lever  having  its  fulcrum  in  the  axis,  and 
the  power,  in  a  state  of  equilibrium,  will  be  to  the  weight,  as 
the  radius  of  the  axle  to  the  radius  of  the  wheel,  or 

Pa=W6.  (119  a) 

But  as  the  circumferences  of  wheels  are  proportioned  to  their 
radii,  the  latter  part  of  the  proportion  may  be  changed  ;  and 
when  the  wheel  and  axle  is  in  equilibrio,  the  power  is  to  the 
weight  as  the  circumference  of  the  axle  to  the  circumference  of 
the  wheel,  or 

P.2ira=W.2*6.  (1196) 

The  simplest  form  of  the  wheel  and  axle  is  represented  be- 
neath. In  it  the  power  is  applied  by  an  endless  rope  passing  over 


the  circumference  of  the  wheel,  and  the  weight  to  a  rope  coiled 
or  wound  around  the  axle.  Such  is  the  form  which  is  habitually 
used  in  our  warehouses. 

It  is  not  necessary  that  the  wheel  should  be  continuous:  a 
single  spoke,  or  several,  projecting  from  the  axle,  will  be  suffi- 
cient. The  axis  may  be  either  horizontal  or  vertical  ;  in  the  for- 
mer case,  an  axle  with  bars  or  spokes,  is  called  a  Windlass  ;  in 
the  latter,  a  Capstan. 

The  windlass  used  in  ships  has  a  number  of  holes  cut  in  the 
direction  of  its  length,  upon  four  different  parts  of  its  circumfe- 
rence ;  handspikes  or  bars  are  placed,  when  the  engine  is  to  be 


Book  ///.] 


MECHANIC    POWERS. 


139 


used,  in  the  row  of  holes  that  is  uppermost,  and  men  springing 
to  these  bars,  act  upon  them  partly  by  their  muscular  force,  ?nd 
partly  by  their  weight.  It  is  therefore  an  application  of  human 
force  that  requires  great  exertion,  and  produces  corresponding 
effects  for  a  short  time. 

The  windlass  which  is  used  by  well-diggers,  has  a  bar  at  each 
end  ;  this  bar  is  bent  at  right  angles,  in  order  to  furnish  a  con- 
venient handle  to  the  persons  that  work  it.  A  handle  thus  formed 
is  called  a  Winch,  and  is  of  frequent  application  in  many  useful 
cases. 

A  capstan  with  a  single  bar,  to  which  a  horse  is  harnessed,  is 
often  used  on  the  shores  of  our  bays  and  rivers  for  the  purpose 
of  drawing  up  logs  and  balks.  In  a  ship's  capstan,  a  number  of 
bars  are  inserted  into  the  head,  in  the  manner  of  the  spokes  of  a 


wheel.  It  is  manoeuvred  by  men  placed  between  these  bars  and 
walking  around.  The  exertion  is  therefore  moderate,  and  can  be 
long  continued.  In  large  ships,  the  capstan  passes  through  the 
decks,  and  thus  a  gang  of  men  may  be  applied  to  it  on  each  deck. 
In  all  these  cases  the  weight  is  applied  to  a  rope  wound  around 
the  axle. 

148.   A  series  of  wheels  and  axles  may  be  combined  together, 
as  in  the  figure,  by  means  of  endless  ropes  or  bands ;  one  of  these 


140 


O7    THE 


[Book  ///. 


is  passed  over  the  circumference  of  a  wheel,  and  the  circumfe- 
rence of  the  axle  of  the  adjacent  wheel ;  the  number  of  bands  in 
the  system  is  always  one  less  than  the  number  of  wheels.  The 
action  is  identical  with  that  of  a  system  of  levers.  The  power, 
therefore,  will  be  in  equilibrio  with  the  weight,  when  the  former 
is  to  the  latter,  as  the  continued  product  of  the  radii  of  all  the  axles 
is  to  the  continued  product  of  the  radii  of  all  the  wheels. 

149.  Wheels  and  axles  may  also  be  combined,  by  making  them 
turn  each  other  by  the  friction  of  their  surfaces,  and  giving  these 
such  a  form  as  to  exert  a  direct  pressure  upon  each  other.  For 
this  purpose — 

(I.)  Projecting  pieces  or  cogs,  as  in  the  figure,  may  be  adapted 


to  the  circumference  of  the  wheel,  and  the  axle  of  the  next  may 
be  formed  of  two  parallel  circular  plates,  united  by  round  staves, 
arranged  in  the  circumference  of  a  circle  :  such  an  arrangement 
is  called  a  Cog-Wheel  and  Trundle  :  or, 

(2.)  The  circumferences  of  both  the  wheels  and  axles  may  be 
cut  into  teeth,  as  in  the  figure  beneath.  Such  a  modification  is 
called  the  Wheel  and  Pinion. 


Book  1IL]  MECHANIC    POWERS.  141 

In  conformity  with  the  principles  of  a  combination  of  levers, 
of  which  this  is  an  obvious  application,  equilibrium  will  exist  in 
a  series  of  wheels  and  pinions,  where  the  power  is  to  the  weight, 
as  the  continued  product  of  the  radii  of  all  the  pinions,  to  the 
continued  product  of  the  radii  of  all  the  wheels. 

Were  the  radii  of  the  pinions,  and  their  number  of  teeth,  exact 
aliquot  parts  of  the  radii  of  the  wheels,  and  of  their  number  of 
teeth,  at  each  revolution  of  the  wheel,  the  same  tooth,  upon  its 
circumference,  would  fall  between  the  same  two  teeth  of  the 
pinion.  From  this  woufd  arise  an  unequal  wear.  To  prevent 
this,  the  number  of  teeth  on  the  wheel  ought  to  be  such  as  is 
prime  to  the  number  of  teeth  qpon  the  pinion.  This  is  usually 
effected  by  adding  one  additional  tooth  to  the  wheels.  A  num- 
ber of  revolutions,  then,  equal  to  the  number  of  teeth  in  the  wheel, 
must  take  place  before  the  same  two  teeth  of  the  adjacent  wheel 
and  'pinion  can  again  come  into  contact.  Such  an  additional 
tooth  is  called  the  Hunting  Cog. 

To  express  the  conditions  of  equilibrium  in  terms  of  the  num- 
ber of  teeth  :  The  power  must  be  to  the  weight  as  the  con- 
tinued product  of  the  number  of  teeth  on  all  the  pinions  is  to  the 
continued  product  of  the  teeth  of  all  the  wheels. 

150.  It  is  of  great  importance  in  systems  of  wheels  and  pi- 
nions, that  the  teeth  should  have  a  proper  curvature,  in  order 
that  the  action  of  the  power  may  be  communicated  to  the  weight 
as  directly,  and  with  as  little  friction  as  is  possible.  It  would 
occupy  too  much  space  to  enter  fully  into  the  detail  of  the  con- 
struction of  the  teeth  of  wheels,  but  it  is  necessary  that  the  gene- 
ral principles  be  explained. 

We  shall  take  the  case  of  two  wheels  situated  in  the  same 
plane. 

Let  two  circles  touch  each  other,  each  being  moveable  around 
the  centre.  A  constant  force  in  the  direction  of  the  common 
tangent  of  the  two  circles,  would  make  the  circumference  of 
each  revolve  with  equal  velocities.  In  order  that  one  of  these 
circles  should  transmit  its  motion  to  the  other,  simple  friction 
might  at  first  be  sufficient,  but  this  would  speedily  wear  them 
away  and  thus  disunite  them.  In  practice,  then,  it  is  necessary, 
that  for  the  two  circles  that  touch  each  other,  two  others  described 
around  the  same  centres,  and  with  diameters  having  the  same 
ratio  to  each  other,  should  be  substituted,  and  that  on  their  cir- 
cumferences should  be  placed  teeth.  These  teeth,  in  order  to  keep 
up  an  equable  communication  of  motion,  must  satisfy  this  con- 
dition :  that  in  the  action  of  the  teeth,  which  will  be  in  the  di- 
rection of  the  common  normal  of  the  surfaces  in  contact,  the  two 
primitive  circles  shall  be  moved  as  if  they  were  propelled  by  a 
force  in  the  direction  of  their  common  tangent. 


149 


OF    TUB 


[Book  III 


Let  there  be  two  circles  whose  radii  are  AB,  BD,  that  touch 
each  other  at  the  point  B :  let  there 
be  fixed  upon  one  of  the  circles  a 
tooth  terminated  by  the  curve  BM, 
and  on  the  other  a  radius  AB,  the 
tooth,  in  turning,  will  move  the  radius 
of  the  other  circle  ;  the  condition  to 
be  fulfilled  is  that  the  curve  BM  shall, 
in  all  its  positions,  touch  the  radius, 
and  the  perpendicular  to  the  radius 
BA,  at  the  point  of  contact,  shall  al- 
ways pass  through  the  fixed  point  B. 
The  curve  which  will  satisfy  the 
conditions,  is  an  epicycloid,  /brmed 
upon  each  of  the  circles  by  a  point  in 
the  other,  supposing  the  latter  to  move 
upon  the  circumference  of  the  former, 
in  the  same  manner  that  a  circle  moves 
upon  a  line  when  the  common  cycloid 
is  generated. 


Thus  in  the  figure,  if  t^o  circle,  whose  radius  is  AB,  be  sup- 


Book  III.]  MECHANIC    POWERS.  143 

posed  to  move  upon  the  circumference  of  the  circle  DBF,  the 
point  F  will  describe  an  epicycloid  DFE,  and  some  portion  of 
this  curve  will  be  the  proper  figure  to  give  to  the  projection  of 
the  teeth. 

The  spaces  between  the  teeth  are  formed  by  supposing  the  ra- 
dius of  either  of  the  circles  to  be  produced ;  and  that  the  point 
on  its  extremity,  shall,  while  the  epicycloid  is  described,  describe 
a  curve  such  as  GHI ;  this  will  give  a  form  proper  for  the  inter- 
val of  the  teeth.  The  two  curves  cannot  be  made  to  unite  except 
at  an  angle  ;  hence  it  is  necessary  to  join  them  by  a  straight  line, 
which  is  a  common  tangent. 

If  the  teeth  of  one  of  the  wheels  have  their  forms  determined, 
the  teeth  of  the  other  will  no  longer  have  the  same  form,  but  must 
be  modified  so  as  to  adapt  themselves  to  the  teeth  of  the  first,  in 
conformity  with  the  condition  we  have  stated. 

Thus  in  the  case  of  the  wheel  and  trundle,  the  staves  of  the 
latter  may  be  considered  as  teeth,  whose  sections  are  circular. 
In  this  case,  the  cavities  between  the  teeth  will  be  portions  of  a 
circle  whose  radius  is  the  same  as  that  of  the  staves  of  the  trun- 
dle ;  the  projections  will  be  curves  parallel  to  the  epicycloid  de- 
scribed upon  the  circle  whose  radius  is  BD,  by  the  circle  whose 
radius  is  AB. 

However  regular  the  curves  may  be,  and  however  completely 
they  may  satisfy  the  prescribed  condition,  there  will  be  an  in- 
equality in  their  pressure  on  each  other.  This  inequality  may 
be  lessened  by  increasing  the  number,  and  lessening  the  size  of 
the  teeth.  In  cases  where  intensity  of  force  is  gained  at  the  ex- 
pense of  velocity,  this  inequality  becomes  less  and  less  percepti- 
ble at  each  addition  of  a  wheel  and  pinion  to  the  system  ;  but  in 
the  case  where  velocity  is  gained,  the  apparent  inequality  is 
multiplied  in  exact  proportion  to  the  increase  of  velocity. 

In  the  former  case,  the  successive  variations  will  be  repeated 
several  times  within  the  time  in  which  two  teeth  are  in  contact; 
in  the  latter,  they  will  be  found  to  affect  several  teelh,  each  of 
which  will  be  unequally  impelled. 

151.  The  circumferences  of  the  wheel  and  pinion  that  are  in 
contact  will  move,  as  has  been  seen,  with  equal  velocities;  their 
moments  of  rotation  are  therefore  proportioned  to  their  respective 
diameters.     Hence,  when  wheels  drive  pinions,  the  intensity  of 
the  force  is  diminished,  and  when  pinions  drive  wheels,  increased. 
In  the  former  case  velocity  is  gained,  in  the  latter  it  is  lost. 

152.  When  a  motion  is  to  be  changed  so  that  its  direction 
shall  lie  in  a  different  plane  from  that  in  which  the  forces  had  before 
acted,  the  wheel  and  pinion  offers  various  modes  of  effecting  the 
change.     Thus  :  the  teeth  may  be  cut  upon  the  surface  of  a  hol- 
low cylinder,  and  stand  at  right  angles  to  the  plane  in  which  the 


144 


OF    THE 


[Book  111. 


wheel  revolves  ;  the  axis  of  the  pinion  must  then  be  perpendicular 
to  the  axis  of  the  wheel,  and  the  motion  will  be  taken  off  at  right 
angles  ;  such  a  wheel  is  called  a  Contrate  Wheel.  The  same  may 
be  effected  by  forming  the  teeth  upon  the  surfaces  of  two  conic 
frusta,  whose  axes  meet,  and  make  with  each  other  an  angle, 
which  is  the  supplement  of  the  required  change  of  motion.  The 
figure  of  the  teeth,  in  this  case,  is  derived  from  a  curve  called  the 
spheric  epicycloid.  Such  wheels  are  called  Bevelled,  or  Mitre 
geering,  according  as  the  angle  their  planes  make  with  each  other, 
is  right  or  oblique. 

Let  VAB,  and  VAD  be  sections  of  right  cones  ;  wheels  and 
pinions  constructed  upon  their  frusta,   will  have  the  same  me- 

FIG.  1. 


chanical  properties  as  if  they  were  in  one  plane.  They  will  be- 
sides change  the  direction  of  motion  to  a  plane,  making  an  angle 
with  that  in  which  the  original  motion  is  performed,  equal  to  the 
complement  of  the  angle  made  by  their  respective  axes,  VPJ,  and 
VC.  In  Fig.  1st,  the  motion  is  taken  off  at  a  right,  in  Fig.  2d, 
at  an  obtuse  angle. 


Book  ///.] 


MECHANIC   POWERS. 


145 


In  the  figure  beneath,  a  pair  of  mitre  wheels,  with  their  teeth 
entering  into  each  other,  is  represented. 


Such  wheels  are  sometimes  equal  in  size,  and  answer  no  other 
purpose  than  to  change  the  direction  of  the  motion. 

153.  In  the  original  and  simpler  forms  of  the  wheel  and  axle,  the 
friction  of  the  pivots  is  estimated  exactly  as  we  have  done  it  in  the 
case  of  the  lever :  but  before  the  machine  can  be  ready  to  be  set  in 
motion,  an  additional  force  must  be  applied  to  overcome  the  resist- 
ance of  the  rope  or  ropes.     In  the  capstan  and  windlass  there  is 
but  one  rope,  and  the  equation  of  final  equilibrium,   when  the 
smallest  force  added  to  the  power,  will  cause  motion  to  begin,  is 

Po=W6+(P+W)i/r+6  (ro+p  W)  ;  (119c) 

in  which  expression,  a  is  the  radius  of  the  wheel,  b  that  of  the 
axle,  and  r  of  the  gudgeon  on  which  the  motions  are  performed. 
Coulomb,  in  order  to  give  an  instance  of  the  application  of  his 
theory,  and  the  results  of  his  experiments  to  practice,  calculates 
the  joint  resistances  of  the  friction,  and  the  rigidity  of  ropes,  in 
the  case  of  raising  8000  Ibs.  by  means  of  a  capstan,  and  finds  that 
one  tenth  part  of  the  moving  power  must  be  expended  upon  these 
resistances. 

154.  In  the  case  of  a  wheel  acting  upon  a  pinion,  or  upon  a 
trundle,  we  shall  not  enter  into  a  full  investigation  of  the  friction  : 
the  old  practical  rule  of  allowing  one  eighteenth  part  of  the  poxver, 
applied  to  its  own  arm  of  the  lever  into  which  the  wheel  may  be 
resolved,  having  been  found  to  correspond  with  the  results  of  the- 
ory.    A  similar  allowance  must  be  made  for  every  increase  in  the 

19 


OJ  THE 


[Book  111 


number  of  wheels  and  pinions  in  the  system  ;  the  additional  fric- 
tion being,  in  each,  one  eighteenth  part  of  the  force  the  wheel 
exerts  upon  the  pinion  it  drives. 

Of  the  Pulley. 

155.  A  Pulley  is  awheel,  moveable  upon  an  axis,  and  having 
a  groove  cut  upon  its  circumference,  over  which  a  cord  passes. 
It  is  enclosed  in  a  box  or  case  that  supports  the  axle,  which  is 
called  its  Block.  The  block  may  be  either  fixed  to  a  firm  sup- 
port, or  moveable.  In  both  cases,  the  power  is  applied  to  one 
end  of  the  rope  ;  in  the  case  of  the  fixed  pulley  No.  1,  the  weight 

No.  1.  No.  2. 


is  applied  to  the  other  extremity  of  the  rope,  in  the  moveable  pul- 
ley No.  2,  the  weight  is  suspended  from  the  box  or  block. 

In  the  fixed  pulley,  No.  1,  the  direction  of  the  motion  is 
alone  changed,  for  the  power  and  weight  have  equal  moments 
of  rotation,  and  hence  may  be  considered  as  acting  upon  the 
equal  arms  DC,  CE,  of  a  lever  of  the  first  kind,  hence 

P=W.  (120) 

In  the  moveable  pulley  No.  2,  the  rope  is  fastened  at  one  end 
to  the  fixed  support,  F  ;  this  may  also  be  considered  as  a  lever, 
but  the  fulcrum  is  not  at  the  centre,  but  at  the  point  D,  hence 

P  :  W  :  :  DC : DE ; 

but  one  of  these  lines  being  the  radius,  the  other  the  diameter  of 
the  same  circle, 

P=*W.  (121) 

In  the  moveable  pulley,  the  direction  of  the  force  P  is  not 
changed.  But  it  is  frequently  desirable  that  the  intensity  of 
the  power  shall  not  only  be  doubled,  but  that  its  direction  shall 


Book  ///.] 


MECHANIC    POWERS. 


147 


be  changed.     In  order  to  effect  this  change,  a  fixed  pulley  may 
be  combined  with  a  moveable  pulley,  as  in  the  figure. 

| — s       Here  the  condition  of  equilibrium  is  siill 

— ^  the  same,  or 


156.  Pullies,  whether  fixed  or  moveable, 
may  be  combined  together'  in  various  man- 
ners ;  thus,  as  in  the  system  beneath  ;  for 
the  weight  that  acts  upon  the  moveable  pulley 
—  -  2  A,  may  be  substituted  a  rope  that 
e  is  wound  around  a  second  pulley, 
B  ;  to  this,  in  like  manner,  a  rope 
passing  over  a  third  pulley,  C,  may 
be  applied,  and  the  weight  may 
be  attached  to  the  box  of  C  ;  the 
pulley,  A.  doubles  the  intensity  of 
the  power,  or 

P=iW; 

the  pulley  B  does  the  same  to  the 
force  which  acts  upon  it,  which  is 
W  and  hence 


The  pulley,  C  produces  the  same 
change  in  the  force  W",  therefore, 

W"=4W; 
whence  we  obtain 

P=|  W. 

It  will  be  obvious,  that  in  such 
a  system,  the  intensity  of  the  pow- 
er is  increased  in  a  geometric  pro- 
gression, whose  common  ratio  is 


14S 


OF    THB 


[Book  Ilf. 


2,  and  whose  number  of  terms  is  the  number  of  moveable  pullies. 
Hence,  in  a  system  whose  number  of  moveable  pullies  is  n, 

2"(P)=W.  (122) 

This  mode  of  combining  pullies,  is  not  convenient  in  practice, 
and  hence,  in  spite  of  its  great  power,  it  is  not  often  used.  A 
system  in  which  the  number  of  fixed  and  moveable  pullies  is 
equal,  and  all  the  moveable  and  all  the  fixed  pullies  are  combined, 
each  kind  in  a  single  block,  although  it  causes  a  less  increase  in 
the  intensity  of  the  power,  is,  on  account  of  its  great  convenience, 
in  much  more  extensive  use.  Such  a  system,  composed  of  three 
fixed  and  three  moveable  pullies,  is  represented  below.  It  will 


Book  ///.]  MECHANIC    POWERS.  149 

be  easily  seen  that  as  the  rope  is  now  continuous,  the  intensity  of 
the  force  cannot  go  on  increasing  in  a  geometric  ratio.  To  deter- 
mine the  condition  of  equilibrium  :  the  weight  is  supported,  in 
this  case,  by  six  ropes,  or  rather  six  separate  parts  of  the. single 
rope;  each  of  them  undergoes  a  tension  due  to  the  force  P;  they, 
by  their,  united  effort,  support  the  weight,  which  is,  therefore, 
the  resultant  of  these  tensions,  or  of  six  equal  and  parallel  forces; 
hence 

P=*W, 

and  for  any  number  (ri)  of  moveable  pullies, 

2n  P=W, 

W 

and  P=^  (123) 

A  similar  system  may  be  formed  by  placing  all  the  pullies 
of  each  kind,  in  a  box  in  such  a  manner,  that  their  axis  may  be 
the  same.  Their  axles,  however,  must  be  separate  ;  for  they  will 
not  all  revolve  in  equal  times.  The  pair,  composed  of  one  fixed 
and  one  moveable  pulley,  nearest  to  the  power,  having,  at  its  cir- 
cumference, the  same  virtual  velocity  with  the  power;  while  the 
pair  nearest  to  the  weight  has  a  virtual  velocity  no  more  than 
twice  that  of  the  weight.  The  velocities  of  the  several  pairs  of 
pullies  will,  therefore,  be  as  the  series  of  natural  nunbers,^  the 
last  term  of  which  is  the  number  of  moveable  pullies. 

Out  of  this  varying  velocity  grows  an  unequal  wear  upon  the 
different  axles,  and  the  evil  would  not  be  diminished  by  making 
the  pullies  revolve  at  the  same  rate  ;  for  this  would  be  in  fact 
impracticable,  without  a  vast  increase  of  friction  growing  out  of 
the  dragging  of  the  rope  over  circumferences  having  naturally 
different  velocities,  but  which  would  be  thus  constrained  to  move 
at  the  same  rate  by  their  connexion. 

These  defects  are  obviated  in  the  blocks  of  White.  In  this, 
each  set  of  pullies  is  turned  out  of  a  single  piece  :  the  concentric 
circles  in  the  figure  on  the  following  page,  are  the  projections  of 
the  pullies,  and  are  in  an  order  of  size  corresponding  lo  the  se- 
ries of  natural  numbers.  Hence  a  common  rotation  upon  the 
same  fixed  axis  may  be  given  to  each  set  of  pullies,  which  will 
have  the  same  velocity  with  the  ropes  that  pass  over  them,  and 
a  single  axle  to  each  block  will  be  sufficient. 


150 


OP    THE 


[Book  III. 


The  modes  in  which  pullies 
may  be  combined,  may  be  varied 
almost  infinitely  :  in  them  all, 
however,  the  same  principles  are 
applicable.  It  is*  not  necessary 
then  that  we  should  pursue  their 
modifications  to  any  greater  ex- 
tent. 

157.  When  the  ropes  are  not 
parallel  to  each  other,  the  direc- 
tion of  the  forces  becomes  oblique, 
and  thus  the  effect  of  a  pulley  or 
system  of  pullies,  will  be  chang- 
ed.    The  action   of  the   power 
and  weight  may,   however,    be 
determined  in  all  cases  where  the 
angles  the  directions  of  the  ropes 
make  with  each  other  is  known, 
by  means  of  the  theorems  of  the 
Composition  and   Resolution   of 
Forces  §  12. 

158.  The  effect  of  friction  up- 
on pullies,  and  of  the  resistance  of 
ropes,   may   be   calculated   upon 
the  principles  that  have  already 
been  laid  down,  in  the  case  of  the 
wheel  and  axle.     In  a  single  pul- 
ley, the  equation  of  the  state  in 
which    the    smallest    additional 
force  will  cause  motion,   is  the 
same  as  in  the  wheel  and  axle, 
except    that    the    radius   of    the 
gudgeon  r  in  (119c)  becomes  the 
same  as  &,  and  the  formula  is 

4-^W).  (124) 

In  a  complex  system  of  pullies,  this  becomes  difficult  of  ap- 
plication. If,  however,  we  assume,  1.  That  all  the  ropes  are 
parallel;  2.  That  all  the  wheels  and  all  their  axles  are  equal  in 
diameter;  3.  That  the  function/,  which  represents  the  propor- 
tion of  the  friction  to  the  pressure,  is  very  small ;  4.  That  the 
resistance  of  the  rope  is  proportioned  to  its  tension,  and  that 
hence,  in  formula  (124)  77»  =  0.  Upon  these  assumptions,  mak- 
ing 7i— the  number  of  moveable  pullies.  and 


Book  ///.] 


MECHANIC    POWERS. 


161 


the  expression  for  the  state  in  which  the  smallest  addition  to  the 
power  will  cause  motion,  may  be  found  to  be 

(125 


"— 1. 


The  friction  of  pullies  has  been  considerably  lessened  by  an 
application  of  friction  wheels,  (see  §  125,)  made  by  the  late  Mr. 
Garnett,  of  Brunswick,  New-Jersey.  Their  plan  is  represented 
beneath. 


The  axle  of  the  pulley,  it  will  be  seen,  rests  upon  six  wheels, 
enclosed  in  a  box,  and  the  friction  is  diminished  in  the  ratio  of 
the  diameters  of  these  wheels  to  the  diameters  of  their  axles. 

Of  the  Wedge. 

159.  The  Wedge  is  a  triangular  prism,  of  some  hard  material, 
whose  section  is,generally  speaking,  isosceles.  The  power  is  applied 
perpendicularly  to  the  surface  on  which  it  acts;  the  weight  is  a 
resistance  which  is  resolved  into  two  parts,  each  of  which  acts 
perpendicularly  upon  the  other  two  faces  of  the  wedge.  In  or- 
der that  equilibrium  shall  exist,  these  forces  must,  §  14,  converge 
to  a  point  within  the  wedge,  and  must  be  to  each  other  in  the 
ratio  of  the  sides  of  the  triangular  section  on  which  they  act. 

It  will  be  at  once  seen  from  §  14,  that  three  oblique  forces 
can  only  be  in  equilibrio  when  they  converge  to  a  point,  and, 
three  such  forces  are  proportioned  in  magnitude  to  the  three 
sides  of  a  triangle  formed  by  lines  drawn  perpendicular  to  the 
direction  of  the  forces. 


1  52  OF    THE  [Book  III. 

160.  In  an  isosceles  wedge,  the  weight  is  applied  to  the  two 
equal  sides,  the  third  side  may  be  called  the  head  of  the  wedge; 
and  equilibrium  will  exist  when  the  power  is  to  the  weight  as 
the  thickness  of  the  head  of  the  wedge  is  to  twice  the  length  of 
either  of  its  sides.  •  This  is  an  obvious  inference  from  the  gene- 
ral proposition,  and  has  the  form  of  the  following  equation,  a 
being  the  thickness  of  the  wedge,  and  b  the  length  of  either  of 
its  sides. 

Wa 
P=-2b'  . 

161.  The  theory  of  the  equilibrium  of  the  wedge  is  of  little 
use  in  practical  mechanics,  for  it  is  generally  used  in  splitting  or 
cleaving,  and  is  rarely,  or  never,  acted  upon  by  the  constant  ap- 
plication of  a  force,  but  is  impelled  by  a  moving  body  striking 
against  it  at  intervals,  or  as  it  is  styled,  by  percussion.     The  ef- 
fects, in  this  case,  are  proportioned  to  the  weight  of  the  moving 
body  multiplied  by  the  square  of  its  velocity.     For  the  aggre- 
gation of  the  particles  of  the  body  into  which  it  is  thus  driven, 
may  be  considered  as  a  constant  retarding  force,  and  from  (616) 

a2 

*=2g: 

it  therefore  appears,  that  the  distance  to  which  a  body,  retarded 
by  a  constant  force,  will  go,  before  it  loses  its  whole  velocity,  is 
proportioned  to  the  square  of  its  velocity  ;  hence  the  striking  body 
will  continue  to  impel  the  wedge,  until  the  latter  has  produced 
an  action  proportioned  to  the  weight  of  the  former,  multiplied  by 
the  square  of  its  velocity. 

The  effect  of  the  wedge  is  still  farther  increased  by  the  agita- 
tion produced  by  collision,  among  the  particles  of  the  body  into 
which  it  isdriven;  they  are  by  thisaction  more  easily  penetrated, 
as  is  obvious  from  the  fact  that  repeated  blows  will  destroy  the 
aggregation  of  the  strongest  substances.  This  total  destruction 
of  aggregation  is,  however,  only  finally  effected  by  numerous 
shocks,  unless  they  be  of  great  intensity  ;  and,  generally  speak- 
ing, the  force  of  aggregation  becomes,  so  soon  as  the  earlier 
blows  cease  to  be  felt,  as  strong,  to  all  appearance,  as  it  was  at 
first.  Hence,  in  the  act  of  splitting  or  cleaving  bodies  by  a 
wedge,  the  body  closes  forcibly  upon  the  wedge  at  the  instant  the 
striking  body  ceases  to  act,  and  by  its  pressure  produces  a  fric- 
tion, sufficiently  great  to  retain  the  wedge  in  the  position  to 
.which  it  has  been  driven.  This  great  friction  adds  another  most 
important  property  to  the  wedge,  namely  :  that,  although  im- 
pelled by  an  intermitting  force,  it  does  not  return  back  to  its 
original  position,  when  that  force  ceases  to  act,  but  retains  all 


- 


MECHANIC    POWERS. 


153 


the  advantage  derived  from  previous  impulses,  to  the  whole  of 
which  any  new  impulse  is  superadded. 

162.  The  valuable  applications  of  the  wedge  are  : 

(1.)  To  all  splitting  and  cleaving  instruments,  and  to  every 
variety  of  edge  tools. 

(2.)  To  obtain  great  pressures  by  means  of  small  forces,  wedges 
being  driven  between  a  firm  obstacle  and  the  body  to  be  com- 
pressed. 

A  printing  press  by  Rust,  of  New- York,  is  also  an  applica- 
tion of  the  wedge.  The  two  rollers,  A  and  B,  fall  into  cavities 


F 

of  the  wedge  CDEF,  and  are  fixed,  one  to  the  upper  part  of  the 
frame,  the  other  to  the  platten.  When  the  press  is  to  be  set  in 
action,  the  small  end,  C.D,  of  the  wedge  is  drawn  out  by  a  com- 
bination of  levers,  the  rollers  then  slide  along  the  plane  faces  of 
the  wedge,  and  a  great  pressure  is  produced. 

(3.)  To  raise  great  weights  to  asmall  height :  each  successive 
impulse  applied  to  the  wedge  raises  the  weight  a  small  distance, 
whence  it  does  not  again  fall  ;  for  the  friction  retains  the  wedge 
in  its  place,  until  a  new  blow  be  struck. 

One  of  the  most  valuable  and  beautiful  applications  of  the 
wedge,  is  to  the  lifting  blocks  of  Seppings,  by  means  of  which 
the  vast  weights  of  the  largest  ships  can  be  raised  and  supported, 
when  placed  in  dock  for  repair.  A  section  of  these  blocks  is  re- 
presented beneath. 


A  modification  of  these  blocks,  which  is  said  to  be  even  more 
convenient  in  use,  has  been  invented  by  Thomas,  an  engineer  in 
the  employ  of  the  American  Navy  Department. 

(4.)  There  is  an  application  of  the  wedge  that  is  called  the 
Lewis,  which  is  employed  in  speedily  attaching  great  weights  to 


20 


. 


1.54 


OP    THS 


[Book  1IL 


the  engines  that  are  employed  to  move  them.  In  a  large  stone, 
for  instance,  a  hole  of  the  figure  of  a  truncated  cone,  that  does  not 
differ  much  from  a  cylinder,  is  cut,  the  larger  base  forming  its 
bottom  ;  into  this  is  dropped  an  instrument  of  the  following  form. 
A  cylinder  is  cut  by  two  planes  equally  inclined  to  a  plane  pass- 
ing through  its  axis,  and  when  the  three  pieces  are  laid  togeth.er, 
the  whole  has  the  exact  cylindrical  form.  A  section  of  this  in- 
strument is  represented  beneath;  the  side  AB  is  placed  lowest, 
and  a  force  applied  to  the  ring  C ;  the  effect  of  this  upon  the 

C 


wedge-formed  piece,  will  be,  to  force  the  outer  pieces  against  the 
aides  of  the  cavity,  and  the  pressure  thus  produced,  will  cause  so 
much  friction  as  to  prevent  the  apparatus  from  being  withdrawn 
by  any  force  not  sufficient  to  break  the  substance  in  which  the 
hole  is  cut.  By  this  simple  and  ingenious  instrument,  stones  of 
several  tons  in  weight  m,ay  be  firmly  and  suddenly  attached  to 
engines  employed  to  raise  them,  and  as  instantly  detached,  when 
their  weight  is  supported,  and  no  longer  acts  upon  the  ring. 

163.  The  principle  of  the  wedge  may  be  applied  to  the  case 
of  pressures  upon  its  triangular  bases  as  well  as  upon  its  inclined 
faces,  and  to  bodies  of  pyramidal  and  conical  figures  :  in  all  of 
these,  the  same  effects  are  produced  by  a  succession  of  blows,  as 
in  the  simple  triangular  prism  ;  and  friction  acts  in  the  same 
manner  to  prevent  their  being  withdrawn.  Of  this  form,  we  find 
applications  in  all  piercing  tools,  and  in  nails,  spikes,  treenails, 
and  other  similar  instruments;  they  are  employed  for  uniting 
the  parts  of  such  bodies  as  permit  them  to  penetrate  without  any 


HI.]  MECHANIC    POWEHS. 

great  difficulty,  when  driven  by  successive  blows,  yet  which 
close  and  retain  them  in  their  place,  by  means  of  great  friction. 

164.  It  will  be  obvious  from  what  has  been  stated  in  speaking 
t)f  the  applications  of  this  mechanic  power,  that  friction,  although 
it  always  resists  the  action  of  any  power  vvhatever,upon  machines 
formed  of  such  materials  as  nature  furnishes,  is  not,  on  that  ac- 
count, an  absolute  loss.      Indeed,   few  of  the  mechanic  powers 
could  act,  were  there  no  friction  ;   the  friction  of  the  ropes,  in 
the  wheel  and  axle,  and  in  the  pulley,  causes  the  cylinders  on 
which  they  rest,  to  turn  ;  and  all  the  most  valuable  applications 
of  the  wedge,  derive  the  principal  part  of  their  usefulness  from 
the  action  of  friction.   So  also  it  will  be  seen,  that  one  of  the  two 
remaining  powers  would  be  of  little  use,  were  it  not  for  its  friction. 

The  parts  of  machines,  of  buildings,  and  other  mechanical 
structures,  could  not  be  held  together  were  it  not  for  friction  ; 
and  we  shall  find  instances  in  which  systems,  that  would  other- 
wise be  in  a  state  of  tottering  equilibrium,  are  rendered  stable  by 
friction. 

165.  The  friction  which  attends  the  use  of  the  most  valuable 
applications  of  the  wedge,  is  not  such  as  can  be  reduced  to  cal- 
culation, nor  indeed  is  it  important  that  it  should. 

Of  the  Inclined  Plane. 

166.  The  Inclined  Plane  is  an  instrument  formed  of  a  plane 
surface,  in  any  position  whatsoever,  except  parallel  or  perpendi- 
cular to  the  horizon.      A   body,   placed  upon  such  a  surface,  is 
actuated  by  its  own  weight,  exerted  in  a  direction  perpendicular 
to  the  horizon,  and  which,  being  a  constant  force,  would  cause  it 
to  descend,  as  shown  in  §  47,   with  uniformly  accelerated  velo- 
city ;  its  descent  is  retarded  by  a  constant  force  consisting  in  the 
resistance  of  the  plane.      These  two  forces,   as  may  be  deduced 
from  §  56,  have  the  ratio  of  the  length  of  the  plane  to  the  length 
of  its  base,  or  of  the  cosine  of  the  plane's  inclination  to  the  hori- 
zon, to  unity,  or 

W 

R= cos-  •• 

The  force  with  which  the  body  tends  to  descend  the  plane,  • 
may  be  represented  by  VV  sin.  i ;  and  as  the  power,  if  applied  in 
a  direction  parallel  to  the  line  in  the  surface  of  the  plane,  in 
which  the  body  would  tend  to  descend,  must,  in  order  to  cause 
equilibrium,  be  equal  to  the  force  with  which  the  body  tends  to 
descend 

P=W  sin.  i.  (127) 


156  or  THE  \Book  III. 

The  sine  of  the  angle  of  inclination  is  the  ratio  between  the  length 
of  the  plane  and  its  height,  hence  : 

167.  In  art  inclined  plane,  the  power,  when  inequilibrio  with 
the  weight,  must  have  to  it  the  ratio  of  the  height  of  the  plane 
to  its  length.      The  inclined  plane,  in  this  case,  may  be  considered 
as  a  wedge  in  equilibrio,  under  the  action  of  three  forces,  the  re- 
lation between  two  of  which,  will  determine  the  conditions  of 
equilibrium. 

But  a  weight  may  be  supported  upon  an  inclined  plane,  when 
the  direction  o-f  the  power  is  not  parallel  to  the  plane.  In  this 
case,  the  power  must  be  resolved  into  two  forces,  one  parallel  to 
the  surface  of  the  plane,  the  other  perpendicular  to  it,  the  latter 
has  no  effect,  and  the  support  \vill  be  wholly  due  to  the  former 
of  the  two  components.  By  §  13,  the  value  of  this  component 
will  be,  calling  the  angle  the  direction  the  power  makes  with 
the  surface  of  the  plane,  a, 

P  cos.  o; 

and  the  condition  of  equilibrium  will  be  represented  by  the  ana- 
logy 

P  cos.  a  :  W  :  :  h  :  /,  (128) 

in  which  h  is  the  height,  and  /  the  length  of  the  plane;  and  there- 
fore, 

P  cos.  o=W  sin.  t. 
If  the  former  act  parallel  to  the  base 

a=«, 
and 

P=W  tan.  t;  (129) 

if  we  call  the  length  of  the  base  b,  we  have 

h 
T=tan.i; 

therefore, 

P:  W  :  :  h  :  b -,   or  (130) 

when  the  power  acts  parallel  to  the  horizontal  base  of  the  plane, 
equilibrium  will  exist  when  the  power  is  to  the  weight  as  the 
height  of  the  plane  to  the  length  of  the  base. 

168.  The  inclined  plane  is  a  mechanic  power  of  a  frequent, 
nay,  of  almost  constant  application,  in  cases  almost  Loo  numerous 
to  be  recited.     Thus:   we  take  advantage  of  natural  slopes  to 
raise  bodies  upon  them,  with  powers  of  less  intensity  than  they 
would  require  if  raised  vertically.      We  lift  weights  into  wheel 
carriages  by  temporary  inclined  planes,  formed  of  parts,  either 
fixed  or  moveable,   with  which   they  are  furnished.    In  raising 


Book  HI.]  MECHANIC    POWERS.  157 

weights  from  cellars,  or  from  one  story  of  a  building  to  another, 
we  convert  the  stairs  into  inclined  planes,  by  laying  skids  upon 
them.  In  all  these  cases,  upon  the  principles  in  §  125,  we  save 
friction  by  making  the  body  roll  instead  of  sliding.  We  may 
also  combine  this  mechanic  power,  in  the  case  of  rolling  bodies, 
with  a  modification  of  the  moveable  pulley  :  thus,  when  a  barrel 
is  to  be  raised  from  a  cellar,  after  laying  skids  upon  the  steps, 
and  thus  converting  them  into  an  inclined  plane,  we  take  a  cou- 
ple of  ropes,  and  making  them  fast  at  the  top  of  the  plane,  pass 
them  around  tho  barrel,  and  bring  them  back  again  to  the  top  of 
the  plane  :  a  force  applied  to  these  ropes  causes  the  barrel  to  re- 
volve like  a  roller,  and  the  intensity  of  the  power,  in  addition  to 
the  gain  it  acquires  from  the  plane,  is  doubled  by  the  action  of 
this  temporary  pulley.  Inclined  planes  are  also  frequently  con- 
structed expressly  for  the  purpose  of  enabling  a  power  to  coun- 
terbalance a  weight  of  greater  intensity  :  thus  in  saw-mills,  the 
logs  are  drawn  to  the  place  where  they  are  to  be  sawn,  upon  a  fix- 
ed inclined  plane.  This  principle  has  also  been  applied  in  the 
cases  of  roads,  railways,  and  canals.  In  Morton's  Marine  Rail- 
way, vessels  of  great  size  are  drawn  from  the  water  upon  an  in- 
clined plane,  by  a  power  exerted  through  thejntervention  of  mo- 
difications and  combinations  of  the  wheel  and  axle. 

169.  The  friction  of  the  inclined  plane  is  easily  determined, 
for  it  is  always  a  function  of  the  pressure  ;  and  the  state  in  which 
the  smallest  addition  to  the  power  will  cause  motion,  may  be 
represented  by  the  formula, 

P=W  sin.  t+/cos.  i.  (131) 

Of  the  Screw. 

170.  The  Screw  is  a  mechanic  power  that  may  be  considered 
to  be  formed,  as  in  the  figure,   by   wrapping  an  inclined  plane 


around  a  cylinder  :  it  is  composed  of  a  spiral  ridge  or  thread  up- 
on the  surface  of  a  cylinder,  which  cuts  every  line  that  can  be 
drawn  upon  its  surface,  and  parallel  to  its  axis,  at  an  equal  angle. 
The  weight  is  applied  to  the  extremity  of  the  cylinder,  and  the 
power  acts  to  turn  the  screw  around  ;  thus  tending  to  propel  the 
weight  in  the  direction  of  the  axis  of  the  screw;  it  therefore  be- 
comes necessary  that  the  screw  shall  move  in  a  cavity  to  which 


153  OP  THK  [Book  III, 

its  thread  adapts  itself,  or  in  a  screw  formed  upon  the  surface  of  a 
hollow  cylinder,  which  the  solid  screw  exactly  fills.  The 
weight  may  be  applied  to  the  circumference  of  the  cylinder,  in 
the  direction  of  a  tangent  to  this  circle.  The  screw,  in  this  case, 
is  an  inclined  plane,  whose  height  is  the  distance  between  the 
convolutions  of  the  thread,  or,  as  we  usually  style  it,  the  distance 
between  the  thread  ;  for  although  there  be  but  a  single  thread 
wound  around  the  cylinder,  yet,  as  when  we  view  it,  we  see  nume- 
rous convolutions,  we  call  each  of  them  a  thread,  as  if  they  were 
actually  separate.  The  base  of  Ihis  inclined  plane  is  the  circumfe- 
rence of  the  screw,  hence  from  §  1 67,  the  power  is  to  the  weight, 
when  in  equilibrio,  as  the  distance  between  the  threads  is  to  the 
whole  circumference  of  the  screw.  It  is  far  more  usual  to  apply 


the  power,  as  in  the  figure,  to  a  lever  at  right  angles  to  the  axis 
of  the  screw,  or  to  a  circular  head,  to  whose  plane  the  axis  of  the 
screw  is  a  normal.  In  this  case  the  condition  of  equilibrium  is, 
that  the  power  be  to  the  weight,  as  the  distance  between  the 
threads  of  the  screw  is  to  the  whole  circumference  described  by 
the  point  to  which  the  power  is  applied. 

Call  the  distance  between  the  threads  d,  the  circumference  of 
the  screw  c  ;  call  the  intensity  of  the  force  exerted  by  the  power 
at  the  circumference  of  the  screw,  W. 

By  the  principle  of  the  wheel  and  axle, 
P  :  W  :  :  c  :  C  ; 
whence 


By  the  principle  just  laid  down  for  the  screw, 
W  :  W  :  :  d  :  c  ; 

Wd 
and  W'  =  --  ; 

whence 

Wd=PC,  \ 

and  P  :  \V  :  :  d  :  C  ;  } 


Book  III.]  MECHANIC  POWERS.  153 

171.  Friction  affects  the  screw  in  the  same  manner  that  it  does 
the  wedge  ;  the  friction  of  the  solid  upon  the  hollow  screw  being 
sufficient  to  prevent  it  from  returning  after  the  power  ceases  to 
act.  Hence  the  screw  may  be  applied  to  many  of  the  purposes 
for  which  the  wedge  is  used.  The  difference  in  their  use  is,  that  in 
the  wedge  the  power  most  frequently  acts  by  a  succession  of  blows  j 
in  the  screw  it  acts  in  the  manner  of  a  constant  force.  Hence 
the  latter  may  be  used  for  raising  great  weights  to  a  small  height, 
and  is  the  most  frequent  instrument  used  for  accumulating  force, 
in  order  to  apply  it  to  pressure.  It  therefore  forms  an  essential 
part  of  the  coining  engine,  the  notaries'  and  printing,  as  well  as 
various  other  presses.  It  is,  also,  like  the  modifications  of  the 
wedge,  employed  for  fastening,  and  holding  bodies  together. 
This  it  does  the  more  effectually,  inasmuch  as  the  screw  must 
either  be  turned  around  and  withdrawn,  by  reversing  the  motion 
by  which  it  entered,  or  forced  out,  by  breaking  its  own  threads, 
or  those  of  the  hollow  screw  formed  to  receive  it  in  the  bodies  to 
be  united.  It  will  therefore  unite  most  bodies  more  effectually 
than  nails  or  spikes,  and  is  alone  applicable  to  the  union  of 
such  hard  bodies  as  cannot  be  penetrated  by  arty  form  of  the 
wedge  ;  such  are,  for  instance,  the  metals,  in  the  mass  of  which 
hollow  screws,  fitted  to  receive  the  solid  screw,  may  be  cut. 

172.  A  screw  may  be  applied  to  the  teeth  of  a  wheel,  and  will, 
by  its  simple  revolution,  cause  the  wheel  to  revolve.  In  this  case, 
the  screw  need  have  no  progressive  motion,  and  is  therefore  form- 
ed upon  a  rod  that  is  free  to  turn,  but  does  not  advance  forwards. 
The  hollow  screw  is  no  part  of  such  a  system,  and  the  machine 
is  a  combination  of  the  screw  with  the  wheel  axle. 


160 


OF    THE 


[Book  in. 

In  such  a  combination,  the  screw  is  called  endless.     It  is  rep- 
resented beneath :  • 


The  power  is  applied  to  the  head  of  the  screw,  the  weight  to 
the  circumference  of  the  axle  :  this  power  will  be  in  equilibrio 
with  a  force  W,  acting  where  the  threads  of  the  screw  tend  to 
turn  the  wheel,  at  whose  circumference,  by  (132), 

PC  W'd 

TV^-g-,  orP=— ; 

and  the  force  W  will  be  in  equilibrio,  with  the  weight  acting 
upon  the  circumference  of  the  axle,  where,  by  (11 9  a), 


hence 


Wdb 


ca 


(133) 

Equilibrium,  therefore,  will  exist  in  the  case  of  the  perpetual 
screw  acting  upon  a  wheel  and  axle,  when  the  power  is  to  the 
weight  as  the  distance  between  the  threads  of  the  screw  multi- 
plied by  the  radius  of  the  axle,  is  to  the  circumference  of  the  head 
of  the  screw  multiplied  by  the  radius  of  the  wheel. 

173.  If  the  solid  screw,  and  the  hollow  one  to  which  it  is  adapt- 
ed, be  of  a  hard  material,  the  motion  cannot  be  free,  unless  the 


Book  ///.]  MECHANIC  POWERS.  161 

intervals  between  their  respective  threads  be  as  equal  as  the  na- 
ture of  materials  will  admit.  As  screws  may  be  cut  with  many 
threads,  within  a  small  space,  they  may  be  applied  to  divide  that 
space  into  as  many  parts  as  there  are  threads  of  the  screw  within 
it,  and  each  of  these  parts  will  be  determined  by  a  complete  re- 
volution of  the  screw  around  its  axis.  But  as  the  screw  may 
have  a  circular  head  adapted  to  it,  and  the  circumference  ot  this 
head  may  be  divided  into  many  equal  parts,  the  division  may  be 
carried  further,  to  the  determination  of  as  many  parts  of  the  inter- 
val between  the  threads,  as  there  are  divisions  upon  the  head  of 
the  screw  ;  for  as  a  complete  revolution  of  the  head  of  the  screw 
corresponds  to  a  progressive  motion  of  the  axis  of  the  cylinder 
on  which  it  is  cut,  through  a  distance  equal  to  the  interval  be- 
tween the  adjacent  threads ;  so  a  partial  revolution  will  corres- 
pond to  a  progressive  motion  in  the  screw,  that  bears  the  same 
ratio  to  the  distance  between  the  threads,  that  this  partial  revo- 
lution bears  to  the  whole  circumference.  In  like  manner,  if  a 
perpetual  screw  be  applied  to  the  teeth  of  a  wheel,  and  both  be 
of  a  hard  material,  the  teeth,  catching  in  succession  in  the  thread 
of  the  same  screw,  must  be  equal  among  themselves.  A  complete 
revolution  of  the  screw  will  move  a  point  upon  the  circumference 
of  the  wheel,  through  a  space  equal  to  the  distance  between  the 
threads,  or  to  the  breadth  of  the  tooth  upon  the  wheel ;  if  a  head  be 
adapted  to  the  screw,  and  its  circumference  divided  into  equal 
parts,  portions  of  the  revolution  of  the  screw  may  be  estimated 
by  means  of  them,  and  these  will  be  the  measure  of  corresponding 
parts  of  the  progressive  motion  of  the  wheel,  through  the  inter- 
val between  the  threads  of  the  screw. 

This  property  of  the  screw  is  applied  to  several  important 
purposes. 

(1.)  Many  of  the  micrometers  which  are  used  in  telescopes, 
for  the  measurement  of  such  angles  as  are  included  within  their 
field,  are  moved  by  means  of  screws;  the  numbers  of  whose  re- 
volutions, and  the  parts  of  a  revolution,  determined  by  the  divi- 
sions of  the  head  of  the  screw,  furnish  the  measure  of  the  angle. 
The  whole  breadth  of  the  field  is  first  determined  by  observing 
the  time  it  takes  an  equatorial  star  to  traverse  the  diameter,  and 
this  time  is  reduced  to  degrees,  minutes,  and  seconds.  To  take  an 
instance:  let  the  circle  ABCD  represent  the  field  of  a  telescope, 
in  which  are  seen  the  horizontal  and  vertical  wires  AC,  BD ; 
the  wire  EF,  is  moved  by  the  micrometer  screw ;  the  breadth  of 


21 


162 


OF    THE 


[Book  111- 


the  field  having  been  determined  by  observation,  it  is  next  as- 
certained by  experiment  how  many  revolutions  and  parts  of  a 
revolution  are  required  to  move  the  wire  across  the  field  ;  and 
the  value  of  each  revolution  and  part  is  calculated  by  a  simple, 
proportion.  The  moveable  wire  is  then  placed  so  as  to  appear  to 
coincide  with  the  vertical  one,  and  one  of  the  objects  being  made 
to  coincide  with  the  latter,  the  screw  is  turned  until  the  moveable 
wire  coincides  with  the  other  object;  then  counting  the  revolutions 
of  the  screw,  and  observing  the  portions  in  excess  upon  the  head 
of  the  screw,  it  will  be  at  once  seen  that  a  measure  of  the  angle 
is  obtained  in  terms  nf  tha  known  divisions  of  the  screw. 

(2.)  A  screw  moving  a  straight  bar  forward,  affords  the  means 
of  dividing  it  equally,  and  to  great  minuteness,  by  means  of  di- 
visions on  the  head  of  the  screw.  Such  is  the  principle  of  the 
Straight-line  Dividing-Engine  of  Ramsden. 

(3.)  A  circular  plate  may  be  made  to  revolve  by  means  of  an 
endless  screw.  A  circular  limb  laid  upon  the  plate,  and  concen- 
tric with  it,  may  be  divided,  by  ascertaining  the  proportion  each 
thread  of  the  screw  bears  to  the  whole  circumference  of  the  cir- 
cle. More  minute  divisions  may  be  obtained  by  dividing  the 
head  of  the  screw.  Such  is  the  principle  of  the  circular-di- 
viding engine  of  Ramsden.  In  a  very  beautiful  application  of 
the  same  principle,  which  is  now  used  by  Mr.  Patten,  an  instru- 
ment maker,  in  New-York,  the  circular  plate  has  360  teeth  on 
its  circumference  :  each  revolution  of  the  screw,  therefore,  cor- 
responds to  a  single  degree,  and  the  more  minute  divisions  are 
obtained  by  divisions  upon  its  head.  For  divisions  of  instruments 
that  do  not  require  much  care,  the  screw  is  turned  by  a  treadle, 
at  each  motion  of  which,  a  cutting  tool  makes  its  mark  upon  the 


Book  III.} 


MECHANIC  LOWERS. 


163 


limb  to  be  divided.  The  same  engine  has  been  applied  to  straight 
lines,  by  placing  upon  the  circular  plate,  a  second  plate  accurately 
centered,  whose  circumference  is  exactly  a  yard;  the  rod  (o  be 
divided  is  moved  forward  by  friction  against  this  plate,  and  con- 
sequently, by  a  complete  revolution  of  the  plate,  through  a  straight 
line  a  yard  in  length  :  ten  revolutions  of  the  screw,  therefore, 
correspond  to  an  inch,  each  revolution  to  the  tenth  of  an  inch, 
and  lesser  divisions  are  obtained  by  means  of  the  divisions  upon 
the  head  of  the  screw. 

(4.)  The  same  principle  was  applied  in  the  astronomical  in- 
struments of  the  last  century,  to  subdivide  the  smallest  divisions 
cut  upon  their  limbs.  As  this  method  has  now  been  superseded 
by  others,  it  requires  no  explanation  in  the  present  treatise. 

174.  It  will  appear  from  reference  to  §172,  that  the  increase 
in  the  intensity  of  the  power,  will,  in  the  screw,  if  the  diameter 
of  its  head,  or  the  length  of  the  bar  applied  to  turn  it,  remain 
constant,  be  in  the  inverse  ratio  of  the  distance  between  the 
threads;  hence,  in  order  that  a  small  power  shall  counterba- 
lance a  weight  of  great  intensity,  the  thread  must  be  very  much 
diminished.  In  this  event,  the  strength  of  the,  thread  may  be  so 
far  lessened  as  to  be  incapable  of  bearing,  without  breaking,  the 
intensity  of  the  resistance.  A  common  screw,  then,  has  a  limit  to 
its  capacity  of  changing  the  intensity  of  the  force,  in  the  strength 
of  the  material  of  which  it  is  constructed.  "Screws,  with  fine 
threads,  have  the  additional  disadvantage,  that,  after  their  work  is 
perfoi  med,  it  requires  a  considerable  time  to  turn  them  back  again 
to  the  point  at  which  their  action  commenced,  in  order  to  apply 
them  to  overcome  a  new  resistance.  Both  of  these  defects  have 
been  remedied  by  a  compound  screw,  invented  by  Dr.  Hunter. 
In  the  figure,  AB  is  a  screw  of  large  thread,  and  consequently  of 


w 


1«4  OF    TBS  [Book  ///. 

little  power  ;  but  it  is  hollow  within,  and  cut  into  threads,  to  re- 
ceive the  second  solid  screw,  BC.  The  threads  of  the  latter  are 
a  lillle  less  than  those  of  the  former.  Were  they  exactly  equal, 
when  the  screw  AB  is  turned,  the  screw  BC  would  enter  into 
its  cavity  just  as  far  as  the  former  is  pressed  forward,  and  the 
point  C  would  remain  at  rest.  But  as  the  threads  of  BC  are 
smaller  than  those  of  AB,  the  former  recedes  during  a  single  re- 
volution of  the  latter,  through  a  space  equal  only  to  its  own 
thread  ;  and  hence  the  point  C  is  pressed  forward,  through  the 
difference  between  the  respective  threads.  The  velocity  of  the 
weight  then  is  no  more  than  the  difference  in  the  velocities  of  the 
screws,  and  by  the  principle  of  virtual  velocities,  which  is  most 
convenient  for  our  purpose,  the  condition  of  equilibrium  is:  the 
power  shall  be  to  the  weight,  as  the  circumference  of  the  circle 
described  by  the  power,  is  to  the  difference  between  the  distances 
of  the  threads  in  the  two  screws.  If  now  there  be  ten  threads  of 
the  outer  screw  in  the  space  of  an  inch,  and  ten  of  the  inner  in 
the  space  of  nine-tenths  of  an  inch,  the  difference  will  be  T47th 
of  an  inch,  and  the  intensity  of  the  power  is  as  much  increased 
as  it  would  be  in  acting  upon  a  screw  having  an  hundred  threads 
within  an  inch;  while  the  strength  of  the  threads  of  the  least 
screw  would  be  more  than  nine  limesas  great,  and  capable,  there- 
fore, of  bearing  more  than  nine  times  the  strain.  It  is  easy  in 
practice  to  make  the  greater  screw  carry  the  lesser  back  with  it 
to  its  primitive  position,  and  hence  it  can  be  placed  in  the  posi- 
tion in  which  it  is  again  t,o  begin  to  act,  in  a  tenth  part  of  the 
time,  in  the  case  we  have  assumed,  and  in  proportion  for  any 
other  difference  between  the  distances  of  the  threads.  A  third 
screw  may  be  placed  within  the  second,  and  the  intensity  of  the 
power  again  increased  upon  the  same  principle. 

175.  The  friction  of  a  simple  screw  depends  not  merely  upon 
the  nature  of  the  materials,  and  their  polish,  but  also  upon  the 
greater  or  less  tightness  with  which  the  solid  and  hollow  screws 
apply  themselves  to  each  other.  It  would  in  consequence  be 
impossible  to  reduce  the  action  of  the  friction  to  any  fixed  law. 
In  the  endless  screw,  when  combined  with  the  wheel  and  axle, 
ii  rmv  be  investigated;  and  the  equation  ofilhe  st.ite  in  which  an 
ad  lition  to  the  p  wer,  however  small,  would  cause  motion,  will 
be  found  to  be,  using  ihe  notation  of  (133) 

p=w-      -  '  (134) 


Book  ///.]  STREHGTH,  &C.  165 

CHAPTER  VII. 

OF  THE  STRENGTH  OF  MATERIALS. 

176.  The  force  with  which  the  particles  of  a  solid  body  resist 
the  forces  that  tend  to  separate  them,  and  which  constitutes  their 
strength,  may  be  found,  in  particular  cases,  by  experiment.  If 
the  experiments  be  multiplied,  both  in  respect  to  the  species  of 
substances,  and  to  the  size  and  circumstances  under  which  por- 
tions of  a  given  substance  arc  employed,  a  general  law  might  be 
finally  obtained,  whence  algebraic  expressions  of  the  relation  be- 
tween the  strength,  the  dimensions,  and  the  material,  might  be 
deduced.  Or  we  may,  by  assuming  an  hypothesis  to  represent 
the  probable  manner  in  which  bodies  are  made  up,  deduce  from 
it  general  formulae,  which,  applied  to  cases  that  occur  in  practice, 
will  be  sufficient,  in  almost  every  instance,  to  represent  the  phe- 
nomena. The  latter  of  these  methods,  although  not  the  most 
accurate,  shall  be  employed  by  us;  but  we  shall  carefully  note 
the  differences  that  exist  between  the  inferences  from  the  hypo- 
thesis, and  the  results  of  experiments. 

177.  Galileo  was  the  author  of  the  hypothesis  that  is  most 
generally  employed  by  writers  on  mechanics,  and  which  will 
suffice  in  all  usual  cases.  He  assumes  that  all  solid  bodies  are 
composed  of  a  great  number  of  parallel  and  equal  fibres,  perfectly 
inflexible  and  inextensible  ;  that  when  they  break,  the  several 
fibres  give  way  in  succession,  and  the  body  turns  upon  the  fibre 
or  fibres  that  are  the  last  to  give  way,  as  upon  a  hinge. 

Leibnitz  observing  the  flexure  that  takes  place  in  bodies  be- 
fore they  break,  assumed  as  the  basis  of  his  hypothesis,  the  fact, 
that  every  body  admits  before  it  breaks,  of  a  certain  degree  of 
extension  :  the  fibres,  therefore,  are  both  extensible  and  flexible; 
and  he  inferred  that  the  strength  of  each  fibre,  instead  of  being 
equal,  varied  with  its  quantity  of  extension,  or  was  proportioned 
to  its  distance  from  the  fixed  point  around  which  the  beam  is  sup- 
posed to  turn. 

The  hypothesis  of  Galileo  has  been  recently  extended  by  Bar- 
low, who  has,  by  introducing  the  circumstance  of  the  occurrence 
of  a  flexure  before  the  beam  breaks,  been  enabled  to  explain  all 
the  cases  which  appeared,  by  a  comparison  of  the  hypothesis 
with  experiment,  to  be  anomalous. 

178.  The  force  which  tends  to  destroy  the  aggregation  of  the 
particles  of  a  solid  body,  may  act  in  four  different  manners. 


16(5  STRENGTH  OF  [Book  HI. 

(I.)  It  may  tend  to  tear  the  body  asunder  by  exerting  an  ac- 
tion in  the  direction  of  its  fibres. 

(2.)  It  may  tend  to  break  the  body  across,  by  a  transverse 
strain,  exerted  either  perpendicularly,  or  obliquely  to  the  direc- 
tion of  its  fibres. 

(3.)   It  may  tend  to  crush  the  body. 

(4.)  It  may  tend  to  separate  the  particles  by  means  of  torsion, 
twisting  or  wrenching  the  body,  by  an  action  in  a  plane  perpen- 
dicular to  its  axis. 

The  resistance  which  bodies  oppose  to  a  force  acting  in  the 
first  of  these  modes,  is  called  the  Absolute  Strength.  In  fibrous 
bodies,  it  is  different,  according  as  the  force  is  applied  in  the  di- 
rection of  the  fibre,  or  at  right  angles  to  it :  for  although  by  hy- 
pothesis we  should  conceive  fibres  to  exist,  even  in  the  transverse 
direction,  yet,  in  nature,  none  such  are  found  ;  and  the  aggre- 
gation is  far  more  easily  destroyed  in  the  latter  case  than  in  the 
former. 

The  resistance  to  a  fracture  across  the  body,  is  called  its  Re- 
spective Strength  ;  and  we  may  call  the  resistance  to  a  twisting 
force,  the  Strength  of  Torsion. 

It  needs  no  reasoning  to  show  that  the  measure  of  the  strength, 
even  in  the  same  body,  will  be  different  in  each  of  these  different 
cases;  and  such  is  the  difference  in  the  manner  in  which  the 
particles  of  bodies  of  different  natures  are  united,  that  there  can 
be  no  general  law  that  will  represent  the  relations  of  these  four 
species  of  resistance.  But  in  conformity  with  our  hypothesis,  it 
will  be  obvious  that,  in  all  the  several  cases,  the  resistance  to 
Jracture  will  vary  with  the  number  of  fibres,  which,  in  homo- 
geneous bodies  will  depend  upon  the  area  of  their  sections.  It 
must  also  depend  upon  the  manner  in  which  the  force  acts  to 
break  the  body. 

Of  the  Absolute  Strength  of  Materials. 

179.  When  the  strain  is  exerted  in  the  direction  of  the  fibres, 
the  force  that  tends  to  break  a  body  will  be  direciiy  opposed  to 
its  force  of  aggregation,  and  the  resistance  must  depend  upon  the 
cohesive  force  of  each  fibre,  and  upon  their  number,  but  upon  no 
other  circumstance. 

To  enable  us  to  express  this  analytically  : 


Book  ///.]  MATERIALS.  167 

Let.  ABCD,  represent  the  area  of  a  prismatic  beam  or  rod  ; 
x,  and  y,  the  ordinate  and  abscissa 
B  of  any  point  of  its  surface ;  8  the  ab- 
solute strength  of  one  of  its  differential 
elements,  abed.  The  force  of  the  ag- 
gregation, may  be  represented  by  the 
weight  which  is  just  sufficient  to  destroy 


d 


it,  which  we  shall  call  U. 

The  area  of  the  section  will  be 
2jydx. 


The  expression  for  the  absolute  strength  will  therefore  be 

V=2afydx,  (135) 

for  the  area  of  each  of  the  elements  will  be  2ydx,  its  strength 
2sydx,  and  the  strength  of  the  whole  will  be  found  by  integrating 
this  expression,  in  which  2s  is  constant. 

In  a  rectangular  bar,  whose  lineal  dimensions  are  a  and  6 
ab=2fydx, 

and 

U=sab,  (136) 

whence 

ab 

•=•0 

.. 
In  a  circle  whose  radius  is  r, 

2fydx=xr2, 
and 

U^rfr2*,  (138) 

whence 

.=-;pr  .  (139) 

180.  When  experiments  are  made  upon  the  resistance  of  rods, 
of  dimensions  given  in  some  conventional  unit  of  lineal  measure, 
to  a  direct  strain,  exerted  by  means  of  loads  estimated  in  some 
conventional  unit  of  weight ;  the  value  of  s  may  be  found  in 
terms  of  the  latter,  and  in  reference  to  an  element  of  the  surface, 
whose  magnitude  is  the  square  of  the  unit  of  length.  The  most 
valuable  experiments  that  we  have  of  this  sort,  are  those  of  Bar- 
low, upon  wood.  These  were  made  upon  cylindrical  rods,  and 
the  strength,  s  of  the  formulae,  deduced  for  an  element  of  the  area 
of  a  square  inch.  The  results  are  contained  in  the  second  of  the 
following  tables.  The  first  table  has  been  compiled  from  various 
other  sources. 


168                                                         STRENGTH   OF  [Book  III. 

TABLE  I. 

ABSOLUTE    STRENGTH    OF    THE    METALS. 

Cast  Steel,  ....  140000  Ibs. 

Gold,  (according  to  Morveau)  80000  Ibs. 

Wrought  Iron,  (Swedish,)  72000  Ibs. 

do             (English,)             -  56000  Ibs. 

do  do  in  the  form  of  chains,  48000  Ibs. 

Bronze,  (Gun  Metal,)  \  .-' .  -  36000  Ibs. 

Wrought  Copper,  -  33000  Ibs. 

Cast  do  -  19000  Ibs. 

Brass,  -  -  17000  Ibs. 

Tin, 4700  Ibs. 

Lead,  -  -  1800  Ibs. 

TABLE  II. 

ABSOLUTE    STRENGTH    OF    DIFFERENT    KINDS  OF    WOOD    DRAWN    IN 
THE    DIRECTION    OF    THEIR   FIBRES. 

Boxwood,           -               "—"  ;    '  -  20000  Ibs. 

Ash,                   -                  -         -  17000  Ibs. 

Teak,       -                            -  15000  Ibs. 

Norway  Fir,      -                  -         -  12000  Ibs. 

Beech,                         -         -         -  11 500  Ibs. 

Canada  Oak,               -         -  11400  Ibs. 

Russia  Fir,        -         -         -  10700  Ibs. 

Pitch  Pine,        -  10400  Ibs. 

English  Oak,                        -  10000  Ibs. 

American  White  Pine,  9900  Ibs. 

Pear  Tree,         -         -         -  9800  Ibs. 

Mahogany,                  -         -  8000  Ibs. 

Elm,          -         -         -  5800  Ibs. 

TABLE  III. 

ABSOLUTE    COHESIVE    STRENGTH    OF    WOOD    DRAWN  IN  A  DIRECTION  i-T 
RIGHT    ANGLES    TO    THE    FIBRES. 

Teak,       ....  818  Ibs. 

American  White  Pine,        -        -  757  Ibs. 

Norway  Fir,      -                           -  648  Ibs. 

Beech,      -                           -         -.  615  Ibs. 

English  Oak,                                 -  598  Ibs. 

Canada  Oak,              -                  -  588  Ibs. 

Pitch  Pine,        -  588  Ibs. 

Elm,         -     H/>"*  509  Ibs. 

Ash,         -     „•>•-•.     -         -         -  359  Ibs. 


Book  ///.]  MATERIALS.  169 

Of  the  Respective  Strength  of  Materials. 

181.  To  apply  the  hypothesis  of  Galileo  to  the  case  of  a  trans- 
verse strain,  we  shall  suppose  in  the  first  place,  the  substance  to 
have  the  form  of  a  prismatic  beam,  that  it  is  firmly  inserted  at  one 
end  into  a  fixed  support,  lies  in  a  horizontal  position,  and  is  acted 
upon  by  a  weight  that  presses  at  the  end  that  is  not  fixed  ;  that 
the  fracture  takes  place  at  the  point  of  support,  beginning  at  the 
upper  side,  on  which  the  weight  presses,  and  terminating  at  the 
other.  The  beam  then,  its  fibres  being  by  hypothesis  inflexible 
and  inextensible,  will  turn  around  the  latter  point  in  a  vertical 
plane.  At  the  instant  of  fracture,  the  two  forces  that  act  are  in 
equilibrio  with  each  other,  their  respective  moments  of  rotation 
must  be  therefore  equal. 

In  the  prismatic  beam,  whose  section  is  ABCD,  the  strength 
_n  will  be  represented  by  the  expression 
(§135),  2s  ydx  ;  but  as  the  effort  of  the 
weight  will  cause  the  beam  to  turn 
around  a  horizontal  axis  passing  through 
the  lowest  point,  the  moment  of  rotation 
of  the  strength  of  each  element,  will  be 
found  by  multiplying  2s  f  ydx  into  the  per- 


pendicular distance  of  its  centre  of  gra- 
vity from  this  axis.  Now  the  mean  of  all  these  distances  will  be 
the  distance  of  the  centre  of  gravity  G,  of  the  whole,  from  the 
point  where  the  fracture  terminates.  The  moment  of  rotation  of  the 
whole  resistance  will  therefore  be,  calling  this  distance  c, 

2csfydx. 
Let  EFHG  represent  the  longitudinal  section  of  the  beam, 


fixed  at  EG  to  a  firm  support,  and  pressed  by  a  weight  acting  at 
F.     Let  the  length  EF=/;  the  moment  of  rotation  of  the 
weight  will  be  I W,  and  at  the  instant  of  breaking, 
lW=2csfydx  ^ 

or  w=^fy^x  [  (140) 

22 


170  STRENGTH  OF  [Book  III. 


In  rectangular  beams,  whose  two  dimensions  are  a  and  6, 
2/  ydx 

c= 
and  the  equation  (140)  becomes 


2/  ydx=ab, 
b 


(141) 
In  square  beams,  whose  side  is  a, 

W=g.  (1416) 

In  cylindric  beams,  whose  radius  is  r, 

or3* 
W=-  -[.  (142) 

In  a  hollow  cylindric  beam,  whose  cavity  is  a  cylinder  having  the 
same  axis  as  the  outer  surface,  the  radii  of  the  two  cylinders 
being  R  and  »•', 


If  the  area  of  the  eylindric  ring  be  equal  to  •n'r2  of  (142) 

W=^;  (144) 

andthe  strength  will  be  to  the  strength  of  the  solid  cylinder  of  (142) 
as  R  :  r,  that  is  to  say  :  In  different  cylindric  beams,  having  the 
same  quantity  of  the  same  material,  and  equal  lengths,  but  having 
different  diameters,  in  consequence  of  cavities  of  greater  or  less 
size  within  them,  the  strengths  are  in  the  direct  ratio  of  their  di- 
ameters. 

We  may  obtain  the  comparative  dimensions  of  solid  and  hollow 
cylinders,  that  will  bear  equal  weights,  by  a  comparison  of  the 
formulae  (142)  and  (143),  whence  we  obtain 

r3  =  R3__R/3 

r=V[R(R'-0].l 

If  the  cavity  of  the  hollow  cylinder  be  not  concentric,  the  strength 
should  increase,  according  to  the  hypothesis,  as  the  cavity  ap- 
proaches the  lower  side,  when  it  is  fixed  at  one  end  only  ;  for  in 
this  case,  the  centre  of  gravity  of  the  section  will  be  further  re- 
moved from  the  point  where  the  fracture  terminates. 

In  a  beam  of  the  shape  of  an  isoceles  triangle,  whose  base  is 
e,  and  altitude  ?, 


if  the  edge  be  placed  uppermost, 


IIL]  MATERIALS.  171 

and 

W=-^;  (146) 

if  the  base  be  placed  uppermost, 
_2t 

c=¥/; 

and 

W~*,  ,(147) 

In  a  square  beam,  whose  diagonal  is  vertical, 


and 


In  a  rectangular  beam,  when  /  and  *  are  constant,  the  strength 
varies  with  a62,  (141).  This  proposition  may  be  applied  to  a  case 
thafmay  sometimes  be  useful  in  practice,  namely  :  To  cut  the 
strongest  possible  beam  out  of  a  given  cylinder.  Thus  a  tree, 
although  a  cone  or  conoid,  may,  for  all  useful  purposes,  be  con- 
sidered as  a  cylinder  ;  for  the  size  of  the  rectangular  beam  that 
can  be  cut  from  it,  will  be  determined  by  the  area  of  its  smaller 
end. 

Let  r  be  the  given  diameter  of  the  cylinder,  *  and  y  the  lineal 
dimensions  of  the  required  beam.  When  this  is  the  strongest 
possible,  xy2  will  be  a  maximum  ; 


xe:yz:r-:  :  1  :  2  :  3.  (149) 

The  cylinder  must  therefore  be  so  cut,  that  the  squares  of  the 
breadth  of  the  beam,  of  its  depth,  and  of  the  diameter  of  the  cylin- 
der, shall  be  in  the  proportion  of  1  :  2  :  3. 


172 


STRENGTH  OF 


[Book  IIL 


This  admits  of  an  easy  geometric  construction.     In  the  circu- 

A 


lar  section  of  the  cylinder  draw  the  diameter  AB,  divide  the  di- 
ameter into  three  equal  parts  by  the  points  F  and  G  ;  from  the 
points  F  and  G,  draw  the  perpendiculars  FC,  GD,  towards  op- 
posite sides,  cutting  the  circle  in  the  points  C  and  D ;  join  AC, 
CB,  BD,  AD  ;  the  parallelogram  ADBC,  will  have  the  required 
property,  and  will  he  the  section  of  the  strongest  beam  that  can 
be  cut  from  the  cylinder,  whose  diameter  is  AB. 
It  will  be  at  once  seen  that 


ABa  :  BC2  :  AC3  :  :  1  :  2  :  3. 


(150) 


1S2.  When  abeam,  lying  in  a  horizontal  position,  rests  upon 
two  props,  and  is  broken  by  a  weight  placed  at  an  equal  distance 
from  the  two  props,  we  may  consider  the  laws  of  its  strength  as 
included  in  the  general  case  of  a  beam  supported  at  one  end  on- 
ly ;  for  if,  according  to  the  hypothesis,  it  break  without  bending, 
we  may  conceive  it  to  be  formed  of  two  beams,  each  inserted  in 
a  firm  support  at  the  place  of  fracture,  and  acted  upon  at  each  end 
by  a  force  equal  to  half  the  weight  that  just  breaks  it,  but  which 
is  directed  upwards  instead  of  downwards.  This  force,  which 
is  equal  to  half  the  weight,  will  act  at  a  distance  which  is  equal 
to  half  the  length  ;  its  effort  is  therefore  equal  to  no  more  than 
a  fourth  part  of  the  effort  of  the  same  weight,  applied  to  the  same 
beam,  if  supported  at  one  end  only;  and  as  this  effort  must  be 
just  equal,  at  the  instant  of  breaking,  to  the  transverse  strength 
of  the  beam,  the  latter  will  be  four  times  as  strong  as  when  sup- 
ported at  one  end  only. 


MATERIALS.  173 

In  the  beam  ABCD,  supported  at  C  and  D,  and  to  which  a 


755- 

weight,  W,  is  applied  at  the  point  E,  which  bisects  AB.  If  we 
suppose  the  half  AEFD,  to  be  firmly  fixed  at  EF,  it  will  be  bro- 
ken by  a  force  applied  at  A,  in  the  vertical  direction,  which 
is  equal  to  \  W  ;  the  moment  of  rotation  of  this  force  will  be  equal 
to  the  respective  strength  of  the  beam,  and  as  the  distance  EA 
is  J/f  (140)  ^ 

—  =2scfydx, 

and 

8scfydx 
W=—^L-  ;  (151) 

In  rectangular  beams,  one  of  whose  faces  is  horizontal,  we  obtain 
in  the  same  manner  as  (141), 

2sab2 
W=  —  .  (152) 

In  square  beams,  (141  6), 

2sa? 
W=—  .  (153) 

In  cylindric  beams,  (142), 


W=—  ;  (154) 

with  a  similar  inference  for  the  case  of  a  concentric  hollow  cylin- 
der, as  in  (143)  and  (144). 

But  if  the  hollow  be  not  concentric,  the  strength  will,  in  the 
present  case,  increase  with  the  approach  of  the  cavity  to  the  up- 
per surface.  And  so  in  triangular  beams,  the  proportions  of  (146) 
and  (147)  will  still  hold  good,  but  they  will  be  stronger  with  the 
edge  uppermost. 

In  a  square  beam,  whose  diagonal  is  vertical,  (148), 

2sa'{  v/2 
w=__  .  (155) 

183.  In  the  case  of  a  beam  lying  horizontally,  and  firmly  fixed 
at  each  end,  the  resistance  will  be  equal  to  that  of  a  single  beam 
supported  by  two  props,  added  to  those  of  two  beams  fixed  at  one 
end  only  ;  for  it  is  obvious,  that  three  fractures  must  be  produ- 
ced, one  in  the  middle,  and  one  at  each  end  ;  at  the  first,  the  re- 


174  STRENGTH  OF  [Book  III. 

sistance  will  be  equal  to  that  of  the  supported  beam,  or  four  times 
as  great  as  that  of  the  same  beam,  if  supported  at  one  end  only. 
The  resistance  in  each  of  the  latter  cases,  will  be  that  of  a  beam 
of  half  the  length  fixed  at  one  end  only,  or  one  fourth  of  the 
last  resistance  ;  the  whole  resistance  will  therefore  be  six  times 
as  great  as  that  of  the  same  beam  when  fixed  at  one  end  only,  or 


(156) 

In  rectangular  beams,  (141)  and  (152), 


In  square  beams,  (141  6)  and  (153), 

3sa' 
W=  —  .  (158) 

In  cylindric  beams,  (142)  and  (154), 

6*1*3 

W=—  .  (159) 

If  experiments  be  made  with  rectangular  beams,  in  either  of  the 
three  several  positions,  the  absolute  strength,  s,  is  determinable 
by  means  of  them,  for  calling  the  weights  that  just  break  them, 
W,  W,  and  W". 

In  beams  firmly  fastened  at  one  end,  from  (141), 
2/W 


In  beams,  supported  at  each  end  on  props,  from  (152) 
IW 


In  beams  firmly  fastened  at  each  end,  from  (157) 
JW" 


184.  When  the  beam,  instead  of  lying  in  a  horizontal  position, 
is  inclined,  an  increase  of  its  strength  takes  place,  which  we  shall 
proceed  to  investigate. 


Book  ///.]  MATERIALS.  175 

Let  either  of  the  beams,  whose  longitudinal  section  is  ABCD, 


lie  in  an  inclined  position,  and  let  the  angle  of  inclination  be  t,  the 
moment  of  rotation  of  the  weight  will  become 

WJ  cos.  i.  (163) 

The  effort  the  weight  exerts  to  break  the  beam,  is  no  longer 
exerted  directly,  but  it  is  unnecessary  to  take  this  obliquity  into 
account,  if  bodies  be  constituted,  as  represented  by  our  hypothe- 
sis ;  for  the  number  of  fibres  that  act  to  resist  fracture,  is  still  the 
same.  In  bodies  that  are  not  fibrous,  as,  for  instance,  in  those  that 
are  crystallized,  the  inference  would  probably  be  different ;  but,  on 
this  point,  we  are  unable  to  refer  to  the  results  of  any  experiments. 
It  will  be  obvious  that  the  same  reasoning  will  be  true,  whether 
the  beam  be  inclined  upwards  or  downwards,  and  is  applicable  to 
the  cases  of  its  being  supported,  or  firmly  fixed  at  both  ends,  as 
well  as  to  that  of  its  being  fixed  at  one  end  only. 

185.  When  the  weight  that  tends  to  break  a  beam  is  not  accu- 
mulated at  a  single  point,  but  is  uniformly  distributed  over  its 
whole  length  ;  its  effort  is  diminished  to  the  half  of  what  it  ex- 
erts, when,  in  the  case  of  a  beam  fixed  at  one  end,  it  acts  at  the 
opposite  extremity  ;  or  when,  in  the  case  of  a  beam  supported  or 
fixed  at  both  ends,  it  acts  in  the  middle.     This  will   be  obvious 
in  the  first  case,  from  the  consideration  that  the  point,  where  the 
weight  acts,  will  be  in  the  middle  of  the  beam,  instead  of  being 
at  its  end;  hence  its  moment  of  rotation  becomes  |  /W;  and  as 
the  other  two  cases  are  deduced  immediately  from  the  first,  the 
same  principle  applies  to  them  also. 

186.  We  have  hitherto  omitted  the  action  of  the  weight  of  the 
beam  itself. 

In  small  beams  indeed,  their  own  weights  are  of  little  impor- 
tance, and  need  hardly  be  taken  into  account  in  the  experiments  ; 
but  in  large  beams  this  is  not  the  case,  as  may  easily  be  seen. 


176  STRENGTH  OF  [Book  HI. 

The  weight  which  breaks  a  beam  is  made  up  of  its  own  weight, 
and  the  weight  which  is  applied  for  the  purpose  :  the  former  acts 
at  the  centre  of  gravity  of  the  beam,  which  in  prismatic  beams  is 
in  the  middle  of  their  length.  Its  moment  of  rotation,  therefore, 
will  be  £  V/,  calling  the  weight  of  the  beam  V  ;  the  joint  effort 
of  the  two  will  therefore  be,  in  the  case  of  a  beam  supported  at 
one  end,  and  placed  horizontally,  and  if  the  additional  weight  act 
at  the  extremity, 

(W+fV)/; 
and  the  formula  (140)  for  the  strength  will  become 

(164) 

In  beams  that  are  similar,  we  may  substitute,  for  2cf  ydx,  the  cube 
of  any  one  of  their  homologous  dimensions  multiplied  by  a  con- 
stant co-efficient  ;  let  then 

fP=2cfydx: 
and  the  above  equation  becomes 

W+$V=sfl*.  (165) 

In  similar  beams  of  the  same  homogeneous  material,  the  weight 
is  a  function  of  the  cube  of  their  homologous  dimension,  as  will 
be  half  the  weight,  or 

d>P=iV, 
and 

W+9/3=s//2.  (166) 

It  will  therefore  be  evident,  that  while  the  strength  of  similar 
beams  increases  only  as  the  square  of  one  of  their  homologous 
dimensions,  the  effort  of  their  own  weight  to  break  them  increases 
with  the  cube  ;  and  thus  a  Hmit  will  be  reached,  when 


The  same  principle  applies  equally  to  the  other  two  cases,  in 
which  beams  are  supported,  or  fixed  at  both  ends. 

187.  We  shall  now  recapitulate  the  results  of  our  hypothesis, 
and  state  what  discrepancies  have  been  observed  between  them, 
and  the  inferences  from  actual  experiment. 

(1.)  In  any  prismatic  beam  whatsoever,  thestrength  is  directly 
proportioned,  to  the  area  of  its  section,  and  to  the  distance  of  its 
centre  of  gravity  from  the  point  where  the  fracture  terminates  ; 
and  inversely,  to  the  length  of  the  beam. 

(2.)  The  strengths  of  beams  lying  in  a  horizontal  position, 
when  fixed  at  one  end  only  ;  when  supported  by  a  prop  at  each 
end  ;  and  when  firmly  fixed  at  both  ends,  are  as  the  numbers 
1:4:6.  That  is  to  say  :  a  beam  firmly  fixed  at  both  ends,  is 
six  times,  a  beam  merely  supported  at  both  ends,  four  times  as 
strong  as  when  it  is  fixed  at  one  end  only. 


Book  ///.]  MATERIALS,  177 

These  several  inferences  from  the  hypothesis,  agree  within  all 
usual  limits,  with  the  results  of  experiments.  The  discrepancies 
are  :  that  the  strengths  increase  in  a  ratio  a  little  greater  than  the 
square  of  the  depth,  in  rectangular  heams  ;  and  decrease  rather 
more  rapidly  than  the  inverse  ratio  of  the  length. 

The  second  of  the  above  propositions  admits  of  the  following 
cases  : 

(a)  In  beams  of  the  same  material,  with  equal  and  similar  sec- 
tions, and  unequal  lengths,  the  strengths  are  inversely  propor- 
tioned to  the  lengths. 

The  lengths  being  equal : 

(6)  In  rectangular  beams  of  the  same  materials,  the  strengths 
are  proportioned  to  the  product  of  the  breadth  by  the  square  of 
the  depth  ; 

(c)  In  square  beams,  the  strengths  are  proportioned  to  the  cubes 
of  the  sides  of  the  square  sections  ; 

(fl]  In  solid  cylindric  beams,  the  strengths  are  proportioned  to 
the  cubes  of  the  radii  ; 

(e)  In  hollow  cylindric  beams,  having  the  same  quantities  of 
material  distributed  around  cylindric  cavities  of  different  diame- 
ters, the  strengths  are  directly  as  the  diameters. 

(3.)  Large  beams  are  weaker  in  proportion  than  small  ones, 
for  their  own  weight  constitutes  a  part  of  the  force  that  tends  to 
break  them  ;  and  in  similar  solid  bodies,  the  stress  growing  out 
of  their  own  weight  increases  as  the  cubes  of  their  homologous 
dimensions,  while  the  strength  only  increases  with  the  squares. 

We  see  from  ihN,  that  models  may  be  strong,  and  capable  of 
bearing  a  stress  far  beyond  any  that  can  be  applied  to  them  ;  yet 
that  machines  constructed  exactly  similar  to  them  in  proportion, 
and  of  like  materials,  but  of  increased  dimensions,  may  become 
too  weak  to  bear  even  their  own  weight ;  that  there  must  be  a 
limit  to  the  size  and  extent  of  any  structures  that  can  be  erected 
by  the  hand  of  man  ;  and  that  a  similar  limit  exists  even  in  the 
works  of  nature.  Thus  in  organic  bodies,  mountains,  hills,  trees, 
the  size  they  can  attain,  without  risk  of  disintegration,  isrestricted 
within  certain  bounds.  In  the  animal  creation,  the  same  princi- 
ple applies,  and  the  limit  is  sooner  reached. 

Our  theory  would  show  that  when  a  body  becomes  weak  in 
consequence  of  an  increase  of  length,  strength  may  at  first 
be  added  by  increasing  its  breadth,  and  still  more  by  increasing 
its  thickness,  the  length  remaining  constant.  Here,  how- 
ever, the  weight  is  increased  in  a  greater  ratio  than  the  length, 
and  finally  becomes  excessive.  The  same  quantity  of  material  may 
assume  a  stronger  form  by  being  fashioned  into  a  hollow  tube  ; 
yet  here  again  a  limit  is  reached  when,  the  circumference  of  the 

23 


178  STRENGTH    OF  [Book  III 

tube  becomes  so  thin  as  to  be  liable  to  be  crushed  by  the  forces 
that  act  upon  it. 

In  animals,  we  find  that  the  smaller  classes  have  bones  far 
more  slender  in  proportion  than  those  of  the  larger  kinds.  Their 
muscles  are  far  less  thick  in  proportion  to  their  length  ;  and  their 
masses  are  diminished  in  a  proportion  much  more  rapid  than  that 
of  the  cubes  of  their  similar  dimensions.  We  find  small  animals 
capable  of  lifting  weights  greater  in  proportion  to  those  of  their 
own  bodies,  than  larger  animals  can  ;  and  in  spiteof  this  additional 
effort,  they  are  enabled  to  continue  their  exertions  for  a  longer 
period  without  fatigue.  To  obtain  the  greatest  possible  strength 
with  the  least  possible  weight,  the  bones  of  animals  have  the 
form  of  hollow  tubes,  as  have  the  quills  and  feathers  of  birds. 
In  the  vegetable  kingdom,  we  find  trees  and  plants  made  up  of 
bundles  of  hollow  tubes ;  and  in  those  where  great  strength  and 
comparative  lightness  are  necessary,  these  are  again  arranged  so 
as  to  form  a  hollow  cylinder;  as,  for  instance,  in  the  whole  family 
of  the  gramina. 

(4.)  When  beams  are  in  an  inclined  position,  their  strength, 
which  we  shall  call  F,  becomes 

F=Wcos.  t.  (168) 

This  deduction  from  the  hypothesis,  is  true  in  practice,  so  long 
as  the  beam  does  not  bend  under  the  effort  of  the  weight  applied 
to  break  it. 

(5.)  When  the  pressure,  instead  of  acting  upon  a  single  point, 
at  the  extremity  of  a  beam  fixed  at  one  end,  or  in  the  middle  of  a 
beam  supported  or  fixed  at  both  ends,  is  equally  distributed  through- 
out the  whole  beam,  twice  the  weight  will  be  required  to  break  it. 

188.  As  far  as  we  have  recited  the  results  of  the  hypothesis, 
they  agree  in  all  useful  cases  with  the  deductions  from  experi- 
ment. But  somp  of  ih^  rules,  deduced  from  the  hypothesis,  do 
not  coincide  with  what  occurs  in  practice. 

Thus  : 

It  has  been  shown  that  there  is  a  difference  in  the  strength  of 
triangular  beams,  according  to  their  position,  with  an  edge  or  a 
face  uppermost;  and  that  this  difference  follows  different  laws, 
according  to  the  manner  in  which  the  beam  is  supported.  Ex- 
periment absolutely  contradicts  this;  for  in  neither  of  the  three 
different  positions  in  which  beams  have  been  tried,  has  any  im- 
portant difference  been  found  in  the  strength  of  a  triangular 
beam,  when  placed  with  an  edge,  or  with  a  face  uppermost. 

So  also,  hollow  cylindric  beams  have  not  been  found  stronger, 
when  the  cavity  has  been  nearer  to  the  side  where  the  fracture 
terminates,  as  they  ought  to  be  in  conformity  with  hypothesis; 
neither  is  a  square  beam  stronger  when  its  diagonal  is  vertical. 


Book  HI.}  MATERIALS.  179 

These  remarkable  discrepancies,  together  with  the  less  impor- 
tant ones  that  have  been  noted  in  the  preceding  section,  arise 
from  an  omission  in  the  hypothesis;  this  does  not  take  into  ac- 
count the  elasticity  and  consequent  flexure  which  materials  un- 
dergo before  they  actually  break.  These  circumstances  have 
been  introduced  into  the  investigation  by  Barlow,  in  his  treatise 
on  "the  Strength  and  Stress  of  Timber;"  to  this  work,  then, 
we  refer  for  the  mode  in  which  the  following  formulae,  that 
coincide  almost  exactly  with  the  results  of  experiment,  have  been 
obtained.  Using  the  same  notation  as  before,  and  calling  the 
angle  of  deflection  d9 

In  a  beam  fixed  at  one  end  and  loaded  at  the  other, 


In  a  beam  supported  at  both  ends,  and  loaded  at  the  middle, 


~  4a62  cos.3  d' 
In  a  beam  fixed  at  each  end,  and  loaded  in  the  middle, 
IW 

~   ' 


In  these  formulae,  s'  differs  froms  in  our  formulae  (160),  (161), 
and  (162),  inasmuch  as  it  is  the  strength  of  the  element  ydx  in 
(140),  (151),  and  (156),  instead  of  being  the  strength  of  2ydx. 
We  have  preferred  our  mode  of  estimating  the  respective  strength, 
inasmuch  as  it  retains  the  connexion  with  the  formula  (139), 
which  gives  the  value  of  the  absolute  strength. 

189.  In  order  to  make  our  formulae  applicable  to  practice,  we 
subjoin  a  table  of  the  respective  strength  of  various  bodies,  re- 
duced to  an  element  of  the  size  of  a  cubic  inch. 

TABLE 

OF    THE    RESPECTIVE    STRENGTH    OF    VARIOUS    SUBSTANCES, 

Metals. 

Wrought  Iron,  Swedish,  -         22000  Ibs. 

do           English,  18000  Ibs. 

Cast  Iron,     -  16000  Ibs. 

Wood. 

Teak,            ....  4900  Ibs. 

Ash,     -                            -  4050  Ibs. 

Canada  Oak,  3500  Ibs. 

English  Oak,  3350  Ibs. 

Pitch  Pine,             -         -         -  3250  Ibs. 


180  STRENGTH  OF  [Book  III. 

Beech,          -  3100  Ibs. 

Norway  Fir,  -         -  2950  Ibs. 

American  White  Pine,     -         -  2200  Ibs. 

Elm,  ....  1013  Ibs. 

We  arc  without  any  good  experiments  on  the  respective 
strength  of  stone.  It  would,  however,  appear  from  some  expe- 
riments of  Gauthey,  that,  in  soft  freestone, 

s=68.7  Ibs. 
In  hard  freestone, 

5=72.75  Ibs. 
And  from  some  experiments  recorded  by  Barlow,  that  in  brick 

5=64  Ibs. 

This  would  make  the  respective  strength  of  stone  and  brick,  far 
beneath  that  of  wood,  or  iron. 

The  preceding  table  may  be  applied  to  the  calculation  of  the 
strength  of  horizontal  beams,  of  any  figure  whatever,  by  the  three 
formulae,  (HO),  (151),  and  (15ti).  These  may,  however,  assume 
a  more  convenient  form  for  practice,  by  calling  the  area  of  the 
beam's  section,  A. 

Let  then  — 


the  area  of  the  transverse  section  of  the  beam  in  sq. 

inches  ; 
c=the  distance  of  the  centre  of  gravity  of  A,  from  the  point  where 

the  fracture  terminates,  in  inches, 
f=the  length  of  the  beam  in  inches, 
s=the  number  from  the  preceding  table. 
W=the  measure  of  the  beam's  strength,  being  the  weight  which 

will  just  break  it  in  pounds. 

In  beams  fixed  at  one  end,  and  loaded  at  the  other, 

W=^.  (172) 

In  beams  supported  at  each  end,  and  loaded  in  the  middle, 

4c5A 
W=-T-.  (173) 

In  beams  fixed  at  each  end,  and  loaded  in  the  middle, 


W=—  -.  (174) 

The  particular  formulae,  for  rectangular  square  and  cylindric 
beams,  will  be  found  at  (141),  (1416),  (142),  (152),  (153),  (154), 
(157),  (158),  and  (159). 


Book  ///.] 


MATERIALS. 


181 


Of  the  Resistance  of  Bodies  to  a  Force  exerted  locvush  them. 


190.  The  resistance  of  bodies  to  forces  that  act  to  crush  them, 
would,  at  first  sight,  appear  to  follow  the  same  law  with  the  ab- 
solute strength  ;  that  is  to  say  :  it  must  be  proportioned  to 
the  area  of  the  body,  and  the  force  of  aggregation  of  its  particles. 
Kxperiment,  however,  shows,  that  the  thickness  of  the  substance 
has  an  important  influence  on  the  pressure  it  is  capable  of  bear- 
ing, without  being  crushed.  In  the  first  place,  it  is  found  that 
very  thin  plates  are  readily  crushed  ;  their  resistance  next  in- 
creases with  the  thickness,  until  it  reach  a  maximum  ;  and  finally, 
decreases  slowly,  with  a  farther  increase  of  thickness.  It  has 
been  attempted  to  frame  a  mathematical  theory,  that  should  re- 
present these  circumstances;  assuming,  that  the  body  was  com- 
posed of  flexible  fibres,  and  that  the  crushing  took  place  in  conse- 
quence of  a  bending  in  the  fibres. 

The  respective  strength,  F,  of  the  pillar,  ABCD,  supposed  to 
be  rectangular,  is  represented  by  the  formula  (141). 


A 


satf 


the  effort  of  the  weight  W,  exerted  to  break  it,  may  be 
represented  by  a  force,  bearing  a  constant  relation  to  the 
weight  and  applied  to  the  middle  of  the  column,  say,  at  the 

WJ 

point  E  ;  its  moment  of  rotation  will  be  -r-  ,  and, 


.=*(—) 

\  21  )  5 


whence, 


and  in  a  square  beam, 


(173) 


(174) 


191.  In  bodies  whose  sections  are  similar,  it  may  be  inferred 
that  the  resistance  to  a  force  exerted  to  crush  them,  ft  propor- 
tioned, directly  to  the  cubes  of  the  homologous  dimensions  of  the 
sections,  and  inversely  to  the  squares  of  their  lengths,  The  most 
complete  set  of  experiments  that  we  have  upon  the  variation  in 
the  strength  of  columns  of  different  lengths,  are  those  of  Mus- 
chenbrook,  and  they  correspond  closely  with  the  above  theory.  The 
subject,  however,  has  not  been  so  fully  tested  as  to  authorize  us 
to  receive  any  theory  with  implicit  confidence.  We  shall,  in 


182  STRENGTH  OF  [Book  III. 

consequence,  give  the  absolute  results  of  experiments,  in  the 
terms  of  the  weights  and  dimensions  of  the  original  record, 
without  attempting  to  reduce  them  to  a  conventional  unit. 

TABLE. 

EXPERIMENTS  MADE  BY  RONDELET,  ON  THE  RESISTANCE  OF  DIFFERENT 
SPECIES  OF  STONE  IN  CUBIC  BLOCKS,  OF  THE  SIZE  OF  5  CENTIMETRES 
IN  EACH  DIMENSION. 

Spec.  Grav.  Crushing  Weight. 

Kilogramme*. 

Swedish  Basalt,      -         -         -  3.065  47.809 

Basalt  of  Auvergne,  No.  1,  3.014  44.250 

do                    No.  2,  2.884  51.945 

do                   No.  3,  2.756  28.858 

Porphyry,      -  2.798  50.021 

Green  Granite,  (Vosges,)  No.  1,  2.854  15.487 

Grey         do      (Brittany,)  2.737  16.353 

Granite,  (Vosges,)  No.  2,  2.664  20.482 

Granite,  (Normandy,)     -  2.662  17.555 

Granite,  (Champ  du  Boul,)      -  2.643  20.441 

Granite,  (Oriental  Rose,)         -  2.662  22.004 

Black  Marble,  (Flanders,)       -  2.721  19.719 

White  Veined  Marble,     -  2.701  7.455 

White  Statuary  Marble,  2.695  8.176 

Experiments  made  at  the  same  time  upon  cubes  of  stone,  whose 
sides  varied  from  3  to  6  centimetres,  showed  that  the  strengths 
varied  nearly  in  proportion  to  the  areas  of  their  bases,  and  were 
not  influenced  by  the  thickness.  This  corresponds  to  the  law  of 
absolute  strength  in  §  179,  and  differs  from  that  we  have  given 
for  the  resistance  to  crushing.  On  the  other  hand,  when  cubes 
of  the  same  size  were  placed  one  upon  another,  a  diminution  in 
the  resistance  was  found  that  does  not  differ  much  from  the  lat- 
ter law. 

By  experiments  made  by  Rennie,  on  the  resistance  of  cast 
iron  to  pressure,  it  was  found  that  the  maximum  strength  was 
reached  at  thicknesses  of  from  ;|ths  to  .',  an  inch,  and  that  a  prism 
a  quarter  of  an  inch  square,  and  half  an  inch  in  depth,  was  crushed 
by  10,000  Ibs. 

A  cube  of  cast  copper,  i  inch  each  way,  was  crushed  by 
731S  Ibs. 

A  cube  of  wrought  copper,  of  the  same  size,  by  6440  Ibs. 

Of  brass,  by  10304  Ibs. 

Of  cast  tin,  by  966  Ibs. 

Of  lead,  by  4S3  Ibs. 

An  inch  cube  of  elm,  is  crushed  by  12S4  Ibs. 

Of  American  pine,  by      -  16C6  Ibs. 


Book  ///.]  MATEHIALS.  183 

Of  Norway  fir,  by  ,-v       -         1928  Ibs. 

Of  English  oak,  by  3860  Ibs. 

Of  the  Strength  of  Torsion. 

192.  By  the  experiments  of  Coulomb,  the  resistance  of  wires 
of  the  same  material,  to  a  force  exerted  to  twist  them,  appears 
to  increase  with  the  fourth  power  of  their  diameters.     A  similar 
result  follows  nearly,  from  experiments  by  Renniei  on  square 
bars  of  cast  iron.      It  would  also  appear,  from  a  simple  course  of 
reasoning,  that  the  resistance  must  diminish  with  the  distance  of 
the  point  in  the  rod  to  which  the  twisting  force  is  applied,  from 
the  place  where  it  is  fixed  ;  for  the  rod  tends,  under  the  action 
of  the  force,    to  form    a  screw,   the   distance    between   whose 
threads  is  the  same  as  the  distance  between  the  two  points.     And 
in  the  inclined  plane  which  the  screw  forms  when  developed, 
the  twisting  force  will  act  as  if  it  tended  to  raise  a  weight  along 
the  surface  of  the  plane,  whose  altitude  is  the  constant  lineal  dimen- 
sion of  the  base  of  the  rod. 

193.  We  have  no  experiments  on  the  absolute  resistance  to 
torsion  :  the  following  are  the  relative  resistances  of  different 
materials  deduced  from  the  experiments  of  Rennie. 

Lead,           -  ...           1000 

Tin,  1438 

Copper,        -  4312 

Brass,           -  4688 

Gun  Metal,  5000 

Swedish  Iron,  -                               9500 

English  Iron,  10125 

Cast  Iron,  10600 

Blister  Steel,  16688 

Sheer  Steel,  17063 

Cast  Steel,  ...         19562 


* 


184  EQUILIBRIUM  OF  [Book  III. 


CHAPTER  VIII. 

OF  THE  EQUILIBRIUM  OF  ARTIFICIAL  STRUCTURES. 

194.  Every  building,  machine,   or  other  nrtificial   structure, 
may  be  considered  as  made  up  of  a  system  of  forces,  and  in  order 
that  it  shall  be  stable,  it  is  necessary,  that  in   this  system,   the 
forces  which  tend  to  overthrow  it  shall  not  exert  an  effort  greater 
than  those  which  tend  to  sustain.     If  equilibrium  exist  among  all 
the  forces  that  act,  the  structure  will   be  stable,   until  some  new 
force  be  applied  to  disturb  this  state  of  equilibrium  ;  but  in  order 
that  it  shall  be  permanent,  the  sustaining  forces  must  have  so  great 
a  preponderance  over  those  which  tend  to  destroy,  that  no  acci- 
dental application  of  extrinsic  force  shall  be  able  to  overcome  the 
equilibrium.     Hence  it  becomes  a  matter  of  great  importance  in 
mechanics,  that  we  should  be  able  to  state  the  conditions  of  equi-> 
librium  that  exist,  among  the  forces  that  are  to  be  found  in  action 
in  those  structures  which  are  of  most  frequent  occurrence. 

195.  When  by  the  action  of  a  disturbing  force,  any  part  of  a 
structure  is  removed  from  its  place,  it  can  only  move  in  one  of 
two  ways;  it  may  be  pushed  directly  forwards  upon  the  base-on 
which  the  structure  rests,  or  upon  the  adjacent  parts  of  the  struc- 
ture itself;  or  it  may  revolve  about  some  fixed  point  or  line.  If 
we  call  the  resultant  of  the  forces,  that  tend  to  support  a  structure, 
its  Strength,  and  that  of  the  forces  that  tend  to  destroy  it,  the 
Stress,  or  Thrust  ;  in  the  former  case,  equilibrium  must  exist  be- 
tween the  strength  and  the  stress  ;  and  in  the  latter,  between  the 
moments  of  rotation  of  these  two  resultants. 

Of  the  Equilibrium  of  Walls. 

196.  A  wall  may  be  considered  as  a  prismatic  structure  ;  and, 
in  its  most  simple  case,  as  symmetric  on  each  side  of  the  vertical 
plane  in  which  the  stress  acts,  and  in  which  its  own  centre  of  gra- 
vity falls.     In  this  case  we  may  leave  the  mass  of  the  solid  itself 
out  of  question,  and  examine  no  more  than  the  conditions  of  equi- 
librium of  its  vertical  section. 


Book  ///.]  ARTIFICIAL  STRUCTURES.  185 

Let  ABCD  be  the  vertical  section  of  the  wall.     Let  the  stress 


S  be  resolved  into  two  forces,  R  and  T,  of  which  R  acts  in  a  ho- 
rizontal, and  T  in  a  vertical  direction,  it  will  be  obvious  that  the 
wall  cannot  be  moved  from  its  place,  except  by  a  progressive  mo- 
tion from  B  towards  A,  or  by  a  rotary  motion  around  the  point 
A. 

The  resistance  to  the  former  of  these,  will  be  the  friction.  The 
pressure  on  the  base,  AB,  will  be  the  sum  of  the  weight  of  the 
wall,  M,  and  the  force,  T,  or  M+T  ;  the  friction  then  will  be 

/(M+T)  ; 
and  the  condition  of  equilibrium  will  be 

R=/(M+T).  (175) 

To  estimate  the  moments  of  rotation  of  the  forces:  from  the 
centre  of  gravity,  g-,  let  fall  a  perpendicular,  g-X,  on  AD  ;  from 
S,  the  point  of  application  of  the  stress,  let  fall  the  perpendicular, 
SE,  on  the  same  line  ;  let  AX— w,  AE=/,  ES=r ;  the  moment 
of  rotation  of  the  weight  will  be  Mm,  of  the  force  T,  T/,  and  of 
the  force  R,  Rr ;  the  two  former  will  concur  to  preserve  the  sta- 
bility, the  latter  to  destroy  it,  and  the  condition  of  equilibrium  will 
be 

Rr=Mm+Tf.  (176) 

If  the  wall  have  a  rectangular  section,  and  be  of  homogeneous 
materials  ;  let  its  height  =a,  its  thickness  =6,  its  density  =  G  ; 
then,  as  the  mass  is  the  product  of  the  bulk  by  the  density, 

M=a6  G.  (177) 

The  distance  of  the  line  of  direction  of  the  centre  of  gravity 
from  the  point  A,  on  which  the  wall  would  tend  to  turn,  will  be 

**• 

The  resistance  to  a  horizontal  strain  will  be 

«6/G;  (178) 

The  moment  of  the  resistance  to  a  rotary  motion  will  be 

i«62G;  (179) 

Therefore,  in  a  rectangular  wall,  the  resistance  to  a  horizontal 
thrust  increases  with  its  thickness  ;  and  the  resistance  to  an  ef- 
fort to  overturn  it,  with  the  square  of  the  thickness. 
24 


ISO  EQUILIBRIUM  O?  [Book  HI. 

197.  In  addition  to  the  action  of  the  stress,  to  move  the  wall 
horizontally,  or  to  overturn  it,  that  part  of  the  stress  which  is 
exerted  in  a  horizontal  direction,  tends  to  break  the  wall  ;  that 
part  which  acts  in  the  vertical  direction,  tends  to  crush  it.  The 
manner  of  the  action  of  a  force  of  the  latter  description  has  been 
explained  in  §  190.  Did  ihe  wall  consist  of  one  piece  of  a  homo- 
geneous substance,  the  resistance  to  the  former  of  these  forces 
may  be  considered  as  a  case  of  a  beam  fixed  at  one  end,  which 
has  been  examined  in  §  181.  But  walls  are  composed  for  the 
most  part  of  separate  portions  of  heavy  substances,  held  in  their 
place,  partly  by  the  friction  of  their  surfaces,  and  partly  by  the 
tenar.it  \-  of  cements  ;  hence,  for  the  quantity  .5,  in  (140),  \ve  must 
sub*ti  i:te  the  value  of  these  resistances.  In  respect  to  both  of 
thes  rii  ciniistanres.  we  have  excellent  experiments  by  Boistard, 
which  are  to  be  found  in  the  Treatise  of  Gauthey  4'//e  la  Con- 
strtic'inn  des  Punlx.  "  By  these  it  appears,  that  the  friction  of 
chis«elled  stores  upon  each  other  is  constantly  four-fifths  of  the 
presMire  ;  that  the  resistance  both  of  mortar  and  water  cement,  is 
proportioned  to  the  surface,  and  is  equal,  in  the  former,  to 
1426j  Ibs.  per  square  foot,  in  the  latter,  to  7561  |bs.  When 
plunged  in  water,  however,  the  strength  of  the  latter  is  rather 
increased  than  lessened,  while  the  former  retains  little  or  no  te- 
nacity. The  friction,  it  is  obvious,  will  be  greatest  in  the  low- 
est joints,  and  in  a  rectangular  wall,  will  decrease  uniformly  from 
the  base  to  the  top. 

19S.  When  a  wall  is  to  resist  a  horizontal  strain,  it  may  be 
strengthened  by  making  one  of  the  faces  sloping,  or  by  building 
buttresses  or  counterforts,  projecting  from  it  at  right  angles.  The 
advantages  derived  from  these  different  methods,  may  be  thus 
investigated. 

(1.)  Let/)  be  the  base  of  the  sloping  part  of  the  wall,  being 
the  addition  to  6  of  (177)  at  the  base,  and  on  the  side  opposite  to 
that  where  the  strain  acts;  the  equation  (175)  will  become 

R=o/G(6+ip);  (180) 

and  the  equation,  (176), 

Rr=aG(^2+6p-fip2)  ;  (181) 

If  the  slope  were  on  the  same  side  with  that  on  which  the  strain 
acts,  the  moment  of  resistance  would  be 

Rr=«G(ifc2+ify>+ip2).  (182) 

This  latter  method  would  therefore  be  less  advantageous  than  the 
former.  Were  the  wall  to  be  increased  uniformly  in  thickness, 
by  the  quantity  £p,  the  moment  of  resistance  would  be 


and  this  would  be  the  least  advantageous  method  of  the  three. 


III.]  ARTIFICIAL  STRUCTURES.  187 

(2.)  Let  the  figure  represent  the  horizontal  plan  of  the  wall 


i- 


Cr 


iv 

with  three  of  its  buttresses,  the  intervals  between  which  are  each 
divided  into  two  equal  parts  in  the  points  G,  H  ;  if  the  horizontal 
strain  act  uniformly  along  the  whole  wall,  its  resultant  will  fall  in 
the  vertical  plane  that  divides  it  into  two  equal  parts ;  and  in  this 
section,  its  centre  of  gravity  will  also  fall ;  let  this  vertical  plane 
be  AB  ;  let  the  height  of  the  wall  =a,  its  breadth  =6  ;  ihe  projec- 
tion of  the  buttress,  AD=c  ;  its  thickness  MN=p  ;  and  the  in- 
terval GD=d ;  the  equation  for  the  resistance  will  become  (175) 
R=afG(bd+cp) ;  (183) 

for  the  moment  of  the  resistance,  (176), 

Kr*aG(|6»d+6e<i+j4>).  (134) 

By  an  investigation  similar  to  that  made  in  the  former  case,  it 
will  be  found,  that  buttresses  applied  to  the  side  of  the  wall  on 
which  the  strain  acts,  are  less  advantageous  than  those  on  the  op- 
posite side,  and  that  either  are  preferable  to  an  addition  of  their 
quantity  of  material  to  a  wall  of  uniform  thickness. 

199.  These  principles  are  applied  to  a  very  great  extent  in 
architecture.     Thus  the  walls  of  the  temples  of  ancient  Ejrypt, 
which  are  the  most  stable  buildings  containing  chambers,  that 
have  been  erected  by  the  hand  of  man,  and  are  pressed  by  heavy 
stone  roofs,  have  a  considerable  external  slope;  the  same  form 
is  given  to  the  terrace  walls  of  fortifications,    which  are  besides 
strengtfiened  within  by  buttresses.      In  the  wonderful  buildings 
of  the  middle  ages,  known  by  the  name  of  Gothic,  the  walls  have 
such  large  and  numerous  apertures  as  almost  to  disappear  en- 
tirely ;  but  in  them,  the  stress  of  massive  vaults  of  stone  is  well 
sustained,  by  means  of  buttresses  of  great  projection,  and  which, 
by  being  built  with  an  external  slope,  unite  the  advantages  of 
both  methods.   As  an  instance  of  this,  may  be  cited  the  buttresses 
of  King's  College  Chapel  at  Cambridge ;  these  project  at  their 
base  more  than  twenty  feet  from  the  body  of  the  building,  and 
the  whole  space  between  them  is  almost  completely  occupied  by 
windows. 

Equilibrium  of  Columns. 

200.  The  reasoning  in  §  1  90  and  §  191,  might  be  considered 
as  directly  applicable  to  the  case  of  columns  having  a  weight  to 


•. 


188  EQUILIBRIUM  OF  [Book  III. 

support  upon  their  summits.  It  is  not,  however,  sufficiently  strict 
to  be  admitted,  when  the  length  is  great  in  proportion  to  the 
area  of  the  section.  A  column,  as  then  stated,  if  its  materials 
be  either  wholly  incompressible,  or  after  they  have  been  compress- 
ed as  far  as  their  elasticity  will  admit,  can  only  give  way,  by  a 
flexure  taking  place  either  in  the.  whole  mass,  or  in  the  fibres  of 
which  it  is  composed.  It  hence  becomes  necessary  to  take  into 
account,  the  mode  in  which  the  stress  acts  to  produce  this  flex- 
ure, and  the  manner  in  which  resistance  of  the  material  opposes 
it. 

Let  us  take  the  case  of  an  elastic  plate  Mwin,  infinitely  thin, 


and  fixed  at  the  point  M  in  such  a  manner,  that  however  it  may 
be  bent,  the  direction  of  the  .tangent,  MT,  shall  remain  constant. 
Let  us  suppose,  that  this^pUte  is  submitted  to  the  action  of  a 
number  of  forces  acting  in  the  same  plane,  which  will  be  that  of 
the  curvature.  Call  the  resultant  of  the  components  of  these 
forces  that  are  parallel  to  the  axis  AY,  P,  and  that  of  components 
parallel  to  AX,  Q ;  the  co-ordinates,  x  and  y,'of  the  curve  being 
referred  to  the  same  two  axes.  It  is  required  to  determine  the 
conditions  of  equilibrium  of  the  forces  P  and  Q,  and  the  elasticity 
of  the  plate  which  will  tend  to  restore  it  to  the  direction  of  the 
tangent  MT. 

If  at  any  point  of  the  plate  in,  the  part  Mn  becomes  fixed,  and 
the  part  mn  perfectly  rigid,  a  case  that  will  not  affect  the  condi- 
tions of  equilibrium  :  the  effect  of  the  forces,  P  and  Q,  will  be  to 
tend  to  turn  the  part  mn  around  the  point  m,  and  the  action  of  the 
elasticity  of  the  plate  will  be  exerted  to  turn  the  same  part  of  the 
plate  in  a  contrary  direction  :  hence  the  elasticity  may  be  con- 
sidered as  a  force  acting  perpendicularly  to  the  line  m/,  and  whose 
intensity,  E,  in  the  case  of  equilibrium,  must  be  equal  to  the  sum 
of  the  moments  of  rotation  of  P  and  Q,  in  respect  to  the  point  m. 
If  p  and  q,  be  the  distances  of  these  forces  from  the  axes  Ax 
and  At/,  their  distances  from  wi,  will  be  p — #,  and  9 — x.  The 
equation  of  equilibrium  will  therefore  be 

P(p— *)  +  Q(9— *)=E.  (185) 

At  any  other  point  than  m,  the  elasticity  of  the  plate  will  act  with 
a  different  intensity.     The  law  usually  assumed  to  represent  the 


Book  ILL] 


ARTIFICIAL  STRUCTURES. 


169 


elasticity  is,  that  the  force  is  proportioned  to  the  tension,  or  is  at 
any  given  point,  inversely  proportioned  to  the  radius  of  curvature. 
If,  then,  E  be  the  value  of  the  elasticity  at  a  point  whose  radius 
of  curvature  is  equal  to  unity,  at  any  other  point  the  elasticity  will 
be 

*>=--  (186) 


i+;6r 


The  well  known  value  of  p  is 


9=' 


dx* 


whence  we  obtain 


E 


(187) 


We  may  consider  this  equation  with  greater  ease,  by  confining 
ourselves  to  the  cases  that  may  occur  in  practice. 

(1).  If  we  suppose  that  the  plate  is  so  situated  that  the  tangent 
MT,  is  parallel  to  the  axis  AX,  and  that  the  force  P  is  the  sole 
force,  and  acts  at  the  end  N  of  the  plate ; 


then  Q=0,  and  if  we  call  the  length  of  the  plate  /, 


(188) 


dx 


If  the  curvature  of  the  plate  be  very  small,  -7-  will  also  become 

dx2 
very  small,  and  its  square  -— -  may  be  neglected  in  the  second 

member  of  the  equation  ;   we  may  take  therefore 

E  &.  ;  (189) 


190  EQUILIBRIUM  OF  [Book  Hi. 

Integrating  we  obtain 

*(Hr>=E£'  <190> 

and  there  is  no  need  of  an  arbitrary  constant  ;  for  when  *=0» 

S=0i 

Integrating  again,  we  obtain 

(191) 


in  which  'k  represents  the  distance  AM. 

If  we  make  the  versed  sine  of  the  curvature  MB=;/*,  when  x=e, 
y=k  —  /;  and  we  obtain  for  the  value  of/,  neglecting  its  algebraic 
sign, 


in  which  we  have  a  relation  between  the  length  of  an  elastic  plate, 
and  the  force  that  is  capable  of  producing  a  given  curvature. 

(2).  If  we  suppose  that  the  force  P  in  its  turn  becomes  equal  to 
0,  and  that  the  force  Q  is  applied  to  the  extremity  N  of  the  plate,  in 


such  a  manner  that  its  prolongation  passes  through  the  point  M, 
at  which  point  the  plate  rests  upon  a  plane  AY,  the  plate  will 

then  be  curved,  as  in  M  m  N.    If  we  again  neglect  -j-j  the  equa- 

"2T 
tion  of  equilibrium  will  become 

Q(9— !/)=E^--  (193) 

Multiplying  by  cfy,  and  integrating  we  have 

J-.2 


In  order  to  determine  the  constant  quantity  A,  let  /  be  the 
greatest  value  of  i/,  and  we  have  at  the  same  time 

!/=9— /? 
and 

-^=0;  therefore 
ay 


Book  ///.]  ARTIFICIAL  STRUCTURES.  101 

and 

A=/3-?', 
substituting  this  value  of  A  in  (193a)  it  becomes 

E-,  (184) 


or 

dy 


dx= 


whose  integral  is 

q—y=fsm.  xV^jr  ;  (195) 

in  which  there  is  no  need  of  an  arbitrary  constant,  for  when 
#=0,  y=q. 

If  we  suppose  that  the  axis  AX  is  identical  with  the  line  MN, 
and  that  the  ys  are  positive,  when  reckoned  downwards  from  this 
line,  then  the  equation  becomes 

y=fSm.xV*jr.  (196) 

In  this  expression  y  becomes  =0,  when  j?=0,  or  #=/;  and 
when  the  curvature  is  very  small,  we  may  consider  /  as  coinci- 
ding with  MN  ;  upon  this  hypothesis  then, 

sin.  JV~  =  0; 

and  this  can  only  be  true  when  />/  vr  is  an  exact  multiple  of  half 

the  circumference  of  a  circle  ;  let  m  be  any  whole  number,  and  #, 
the  relation  of  the  circumference  of  a  circle  to  its  diameter  ;  we 
can  express  this  fact  as  under 

/A-—  • 
VE~  2    ' 

from  which  we  obtain  for  the  value  of  Q, 


and  the  equation  (196)  takes  the  form 

Wl*flf 

y=/sin.  —t  x.  (198) 

When  m=l  which  is  the  smallest  possible  value, 

\  ^2£ 

Q=  4^   ,  (199) 

and 

y=/flin-  ~  x  ;  (200) 


EQUILIBRIUM  OF  [Book  If  I. 

from  these  we  may  deduce  that  when  the  value  of  Q  is  less  than 
in  (199),  it  will  not  produce  any  effect  on  the  elastic  plate. 

This  last  case  will  represent  the  action  of  a  weight  W,  upon  an 
elastic  plate,  standing  vertically  upon  a  horizontal  plane.  We 
have  for  the  value  of  the  weight  that  will  produce  no  flexure, 

W=^  •  (201) 

If  instead  of  a  single  elastic  plate  of  infinite  thinness,  we  con- 
sider the  forces  to  act  upon  a  body  of  definite  magnitude  ;  let  the 
forces  be  as  before,  P  and  Q,  acting  in  the  plane  XY, 


parallel  to  the  axes  AX  AY,  let'Mmn  be  the  curvature  which  the 
action  of  the  forces  has  given  to  one  of  the  generating  lines  of  the 
surface  of  the  body,  and  suppose  that  through  some  point  m,  of  this 
curve  a  plane  passes,  to  which  the  curve  at  that  point  is  a  nor- 
mal, and  that  its  intersection  with  the  body  be  represented  by  a 
curve  such  as  mm'.  Refer  this  curve  to  two  rectangular  axes  aw, 
and  at",  that  lie  in  its  plane,  of  which  aw  is  perpendicular  to  the 
plane  in  which  the  forces  P  and  Q  act,  and  touches  the  curve  mm 
on  the  side  where  the  action  of  these  forces  tends  to  bend  the 
body  ;  call  the  co-ordinates  of  this  curve  w  and  v. 

The  body  may  be  considered  as  made  up  of  an  infinite  num- 
ber of  elastic  fibres  ;  and  if  we  suppose,  as  before,  that  the  part 
M»  becomes  fixed,  and  the  part  mn  inflexible,  we  shall  have  again 
for  the  condition  of  equilibrium,  an  equality  between  the  moments 
of  rotation  of  P  and  Q,  and  the  force  of  elasticity.  The  latter 

E 
for  any  one  fibre,  whose  base  is  dw  x  dv  is  by  (186),  —  dw  dv  ; 

E 

its  moment  in  respect  to  the  axis  aw  is  —  dw  vdv.     The  moment  of 

F 

rotation  of  the  whole  base  mm'  will  be  —  ffdw  vdv,  and  the  con- 
dition of  equilibrium  (186), 

p(p-.x)  +  Q(q—y)  =  -ffdw  vdv.  (202) 

If  the  section  of  the  body  were  a  rectangle,  one  of  whose  sides 


HI.]  ARTIFICIAL  STRUCTURES.  193 

corresponds  to  av, 

ffdw  vdv=^  .  (203) 

If  the  section  were  a  circle,  whose  radius  is  r, 

ffwdv  dv=«r*  .  (204) 

If  the  body  were  a  cylindric  tube, 

ffdw  fldr=,r(R3—  d»'2),  (205) 

and  the  relation  between  the  diameters  of  a  solid  cylinder,  and  a 
cylindric  tube,  of  equal  strengths,  would  be 


It  will  be  therefore  obvious,  by  comparing  the  four  last  equations 
with  (141),  (142),  (143),  (144),  and  (145),  that  the  resistance 
to  flexure  follows  the  same  laws  as  the  resistance  to  fracture,  so 
far  as  the  strength  has  reference  to  the  area  of  the  section  ;  but 
as  respects  the  length,  it  will  be  seen  by  reference  to  (197),  and 
(199),  that  the  strength  to  resist  flexure,  diminishes  with  the  in- 
crease of  the  square  of  that  dimension. 

To  apply  this  to  the  case  of  a  column.   Suppose  a  solid  to  be 


placed  vertically — that  it  is  charged  with  a  weight  W  at  its  up- 
per extremity,  and  that  all  its  horizontal  sections  are  circular.  In 
this  case,  W  =  Q,  and  if  we  take  for  the  origiri  of  the  co-ordinates 


the  point  that  bisects  the  height,  f=q  ;  the  first  term  in  (202) 
may  be  neglected.  Combining  (202)  and  (204),  we  obtain  for 
the  equation  of  equilibrium, 

E 

:  —  1JT3  .  (207) 


The  smallest  quantity  of  material,  or  the  greatest  strength  with 
a  given  quantity  will  be  obtained,  when  the  resistance  to  flexure 
is  the  same  in  every  section.  In  this  case  the  radius  of  curvature 
of  the  flexure  the  beam  undergoes,  is  constant,  and  the  curve  an 
arc  of  a  circle,  whose  radius  is  £. 

In  a  solid  of  equal  resistance,  the  curve  assumed  in  bending  will 
have  a  constant  radius  of  curvature,  or  be  a  circle  whose  radius  is 
p.  The  value  of  y  will  become 


substituting  this  value  of?/  in  the  preceding  equation,  we  obtain 


This  will  be  a  maximum  when  x=Q,  or  the  column  must  be 
thickest  in  the  middle  of  its  length. 

Were  this  analysis  carried  farther,  it  might  be  inferred  that 
the  column  should  diminish  towards  each  end,  to  a  point,  and 
should  therefore  have  the  figure  of  a  spindle. 

In  the  foregoing  investigation,  however,  it  has  not  been  taken 

25 


194  EQUILIBRIUM  OF  [Book  III. 

into  view,  that  the  resistance  to  a  crushing  force  depends  upon 
the  number  of  fibres,  and  consequently  in  a  homogeneous  body, 
on  the  area  of  the  surface  pressed.  The  above  inference  will, 
therefore,  only  hold  good  in  the  impossible  case,  that  all  the  fibres 
should  be  curved,  and  meet  at  the  two  extremities  in  a  point. 

Let  us  next  take  the  case  of  a  column  resting  on  a  horizontal 
base,  and  of  a  circular  section  throughout,  formed  by  the  revolution 
of  a  curve  around  the  axis  ;  and  that  a  force  acts,  which  is  distri- 
buted in  the  ratio  of  the  external  surfacesof  the  horizontal  sections, 
and  whose  directions  are  all  parallel  to  each  other,  and  perpendi- 
cular to  the  axis. 

Call  the  resultant  of  these  forces  R.  It  is  obviously  an  instance 
of  the  case  No.  (1),  of  our  preliminary  analysis. 

The  equation  of  equilibrium  obtained  from  (202)  and  (203)  is 

Rx=—«r*  . 

P 

If  n  be  the  pressure  upon  the  unit  of  the  surface,  we  have 

E 


whence  we  obtain 

—Sh"  (209) 

the  generating  curve  is,  therefore,  one  convex  towards  its  axis, 
and  in  which  the  abscissas  are  proportioned  to  the  squares  of  the 
ordinates. 

201.  From  the  preceding  course  of  reasoning  it  may  be  in- 
ferred : 

That  the  strength  of  a  column  is  proportioned  directly  to  its 
area,  and  inversely  to  the  square  of  its  length;  that,  were  its  fibres 
capable  of  being  collected  at  each  end  into  a  point,  it  ou^ht  to 
have  the  form  of  a  spindle.  As  this,  however,  is  not  the  case, 
it  may  be  inferred,  that,  were  reference  to  be  had  only  to  the 
weight  the  column  is  to  support,  it  ought  to  be  somewhat  thicker 
in  the  middle  of  its  length.  As  it.  has  also  its  own  weight  to  sup- 
port, the  swell  ought  to  lie  lower  than  half  the  height,  for  in  this 
way  the  centre  of  gravity  will  be  lowered,  and  the  base  enlarged  ; 
and  thus  the  resistance  to  a  force  exerted  to  overturn  it,  or  to  one 
tending  to  thrust  it  horizontally  from  its  place,  will  be  also  increa- 
sed. 

The  practice,  then,  of  architects  is  founded  on  reason,  and  is  as 
follows  :  In  the  Grecian  Doric,  the  columns  are  truncated  cones, 
whose  least  base  is  uppermost.  In  the  Roman  Doric,  in  the  Ionic, 
the  Corinthian,  and  Composite  orders,  the  columns  are  sometimes 
made  cylindrical  for  one  third  of  their  height  from  the  base,  and 

•    ' 


Book  HI.]  ARTIFICIAL  STRUCTURES.  195 

for  the  remaining  two  thirds  of  their  height,  are  frusta  of  cones  ; 
at  other  times  they  swell  at  one  third  of  Iheir  height,  the  whole 
external  surface  being  one  of  revolution,  formed  by  a  curve  con- 
cave towards  the  axis.  It  is  most  usual  to  make  this  curve  a 
conchoid. 

202.  When  the  weight  exerts  a  lateral  thrust,  the  upper  and 
lower  surfaces  of  the  column  remaining  constant,  it  may  be  infer- 
red from  §  1 98,  that  the  axis  of  the  column  ought  not  to  be  vertical, 
but  should  be  slightly  inclined,  towards  the  side  against  which 
the  weight  acts.     By  a  recent  examination  of  the  columns  of  the 
Parthenon,  it  hns  been  found  that  this  principle  is  there  applied, 
and  to  great  advantage.     The  columns  of  the  external  perystyle 
in  this  perfect  specimen  of  Grecian  architecture,  all  have  their 
axes  slightly  inclined  inwards. 

203.  When  a  column  has  no  weight  other  than  its  own  to  sup- 
port, but  is  acted  upon  by  a  stress  exerted  perpendicular  to  the 
axis,  the  form  should  be  that  of  a  conoid,  formed  by  the  revolu- 
tion of  a.  curve  convex  towards  the  axis.      And  which,  when  the 
stress  is  proportioned  to  the  surface,  as  is  the  case  in  the  action  of 
a  fluid,   should   be  a  curve  whose  abscissas  are  in  arithmetic, 
and  ordinates  in  geometric  progression.     We' have  instances  of 
the  application  of  this  principle  in  the  works  of  nature,  in  the 
manner  in  which  the  trunks  of  tree  rises  from  their  roots,  and 
their  branches  from  their  insertion  into  the  trunk,  or  into  larger 
branches.    In  the  arts,  we  find  a  beautiful  application  of  it,  by 
Smeaton,  in  the  construction  of  the  Edystone  Lighthouse.   In  this 
remarkable  edifice,  obstacles  of  the  most  appalling  character  were 
to  be  overcome  in  its  erection  ;  and  it  is  frequently  exposed  to 
violent  swells  of  the  sea.     The  rock  on  which  it  was  built  is  in- 
clined :  advantage  was  taken  of  this  circumstance  to  cut  it  into 
steps,  to  each  of  which  one  of  the  lower  courses  of  stones  is  adapted. 
These  steps  are  so  formed,  that  one  at  least  of  the  stones  of  each 
course  is  dovetailed  to  the   rock  ;   the  remaining  stones  are  so 
cut,  that  none  of  them  can  be  removed,  without  being  lifted,  unless 
the  dovetail  should  be  disunited  ;  and  throughout  all  the  courses, 
no  one  of  the  outer  stones  can  be  removed  horizontally   without 
moving  all  those  in  the  same  course.  In  addition  to  the  tenacity  of 
the  cement,  the  courses  were  connected  by  dowels  of  the  form  of 
cubes,  of  a  hard  stone,  one  half  of  each  of  which  is  inserted  into 
each  course.  This  structure  has  now  stood  in  its  perilous  situation 
for  seventy  years,  and  has  borne  without  injury,  the  great  stress  to 
which  it  is  exposed.   A  structure  of  similar  character  has  more  re- 
cently been  erected  on  the  Bell  Rock,  at  the  mouth  of  the  Firth 
of  Forth,  in  Scotland. 


198  EQUILIBRIUM  or  [BooAf ///. 

Equilibrium  of  Terraces. 

204.  When  earth  is  piled  up  into  mounds  or  terraces,  we  may 
conceive  that  the  faces  of  these  were  at  first  vertical  ;  the  earth 
composing  the  upper  part  of  the  face,  being  loose,  will  separate 
itself  and  fall;  and  thus  the  base  will  be  gradually  enlarged,  and 
the  face  become  more  and  more  inclined,  until  the  friction  on 
the  surface  of  the  inclined  plane,  thus  formed,  becomes  equal  to 
the  force  with  which  the  earth  would  tend  to  descend  it.  At  this 
point,  any  farther  wear  of  the  mound  would  cease,  and  it  would 
become  stable,  each  particle  on  its  surface  being  in  equilibrio,  un- 
der the  action  of  its  own  weight,  the  resistance  of  the  surface  of 
which  it  forms  a  part,  and  the  retarding  force  of  friction. 

In  looie  soils,  the  earth  will  not  be  supported  until  the  surface 
make  an  angle  of  60°  with  the  vertical  plane  ;  in  tenacious  soils, 
the  support  may  take  place  when  the  angle  reaches  54°. 

205.  If  it  be  required  to  support  a  terrace  by  a  vertical  wall,  it 
must  be  so  constructed  that  it  shall  be  able  to  bear  the  horizontal 
thrustof  the  prismatic  massofearth,  which  lies  above  the  plane, that 
would  form  the  surface  of  a  bank  that  would  be  itself  supported. 
But  this  prism  is  itself  partially  supported  by  friction,  and  we  must, 
before  we  can  ascertain  its  horizontal  thrust,  ascertain  how  much 
of  the  force  it  would  exert  in  a  horizontal  direction,  is  counterac- 
ted by  the  friction. 

Let  us  suppose  a  weight,  W,  to  lie  upon  a  plane  inclined  to  the 
horizon  at  an  angle  p ;  and  that  a  force,  A,  is  applied  in  a  horizon- 
tal direction,  which,  with  the  friction,  just  supports  the  weight.  If 
we  resolve  each  of  the  forces  R  and  A,  into  two  others  ;  one  of 
which  is  parallel,  the  other  perpendicular  to  the  plane,  they 
become 

R  cos.  fx,  R  sin.  /m  ; 

and 

A  sin.  fj.,          R  cos.  p. 

The  two  which  are  parallel  to  the  plane,  act  in  contrary  directions  ; 
their  sum  is,  therefore, 

R  cos.  j& — A  sin.  fx. 
The  other  two  concur  in  direction  ;  their  sum,  therefore,  is 

R  sin.  j^-f-A  cos.  p, 

and  represents  the  whole  pressure  on  the  plane ;  the  friction 
will,  therefore,  be 

/(Rsin.  fx)H-/(Acos.  <x)  ; 

and  as  the  friction  is  equal  to  a  force  that  will  just  support  the 
weight  upon  the  plane,  this  value  of  the  friction  will  be  equal  to 
that  of  the  forces  that  act  parallel  to  the  plane.  Forming  an  equa- 


Book  ILL] 


ARTIFICIAL  STRUCTURES. 


197 


tion  of  these  two  expressions,  we  obtain  from  it  for  the  value  of 

.  (210) 


tan. 


If,  than,  the  earth  be  supported  by  a  wall,  the  second  member  of 
this  equation  will  represent  the  horizontal  thrust  of  the  earth. 

Let  us  now  apply  this  to  the  investigation  of  the  horizontal  thrust 
of  the  prism,  BCE. 


Let  the  angle  CBE=fji ; 

the  line  BC^a; 

and  the  variable  ordinate  in  the  direction  of  BC=x. 
The  element  of  the  surface  will  be 
xdx  tan.  /x  ; 
and  if  g  be  the  density  of  the  earth,  its  weight  will  be 

gxdx  tan.  p  ; 
the  horizontal  thrust  of  the  element  will  be 


or 


tan. 


If  we  make 


1— /tan.  p  \ 


') 

cot.  jx  / 

1— /tan.  p  _ 

the  thrust  of  the  element  is 

WLgxdx. 

Integrating,  and  making  #=a,  we  obtain  for  the  value  of  the  whole 
thrust 


(211) 


The  moment  of  the  thrust  of  the  element  will  be 
M.gxdx(a — x). 

Integrating,  and  making  x=a,  we  have  for  the  moment  of  the 
whole  thrust 


198  EQUILIBRIUM  OF  [Book  ///. 

The  expression,  (210),  will  become   =0,  when  tan.  jx=0,  or 
when  tan.  f/,=     •  between  these  two  limits  there  will  be  a  value 

when  the  thrust  will  be  a  maximum,  determined  by  making 
dM=0  ;  from  this  we  obtain 


if  we  substitute  this  value  in  (210),  we  obtain  for  the  thrust, 

ia°-tan.af*; 
and  for  the  moment  of  the  thrust, 

£a3«-tan.  V 

The  angle  whose  tangent  is  —  /-f  V  (I—  /2),  is  just  double  of 
that  whose  tangent  is  _,  and  the  latter  angle  is  that  at  which  the 

earth  would  just  be  supported  by  its  own  friction  ;  in  loose  earth, 
then, 

fx=30°,  and  tan.  fx==>/i  ; 
in  tenacious  earth, 

(A  =27°,  and  tan.  ^=  V  ^  nearly. 
Hence,  in  the  first,  the  thrust  will  be 

*a»ff>  (212) 

and  its  momentum, 

rV^-  (213) 

In  the  second,  the  thrust  will  be 


and  its  momentum, 

jfa>g.  (215) 

It  has  been  already  seen,  (179),  that  the  moment  of  the  resist- 
ance of  a  wall,  whose  altitude  is  a,  to  a  horizontal  thrust,  is 


and  this,  in  the  case  of  equilibrium,  must  be  exactly  equal  to  the 
moment  of  the  thrust,  or  in  the  case  of  loose  earth, 


(216) 
whence  we  obtain  for  the  thickness  of  a  rectangular  wall, 

fe=j  ^(aVG).  (217) 

The  value  of  the  thickness  of  a  wall  with  any  given  slope,  may 
be  in  like  manner  obtained  from  (181),  and  that  of  a  wall  with 
buttresses  from  (184). 

Of  the  Equilibrium  of  Arches. 

206.  When  an  aperture  of  considerable  extent  is  to  be  covered 
by  a  mass  of  any  material  whatsoever,  it  will  appear  at  once, 
from  what  has  been  said  in  §  186,  that  there  is  a  limit  to  the  use 
of  beams  or  horizontal  lintels,  growing  out  of  the  difference  be- 


Book  ///.]  ARTIFICIAL  STRUCTURES,  199 

tween  the  ratios,  in  which  the  respective  strength,  and  the  action 
of  their  own  weight  to  break  them,  increase.  This  limit  will  be 
reached  earlier  in  stone  than  in  any  other  material  of  which  we  have 
treated,  inasmuch  as  its  respective  strength  is  but  small,  while 
its  weight  is  great.  So  also,  in  this  material,  it  is  frequently  dif- 
ficult to  obtain  pieces  sufficiently  large  to  form  lintels,  even  of  a 
size  within  the  limit  at  which  they  would  become  too  weak.  In 
all  these  cases,  we  have  recourse  to  what  is  called  an  arch. 

An  arch  differs  from  a  lintel,  inasmuch  as  it  is  composed  of  a 
number  of  pieces  of  the  material,  arranged  in  such  a  u.anuer  as 
mutually  to  sustain  each  other;  and  as  each  piece  has  but  small 
dimensions,  measured  in  a  horizontal  direction,  each  will  sustain 
a  considerable  vertical  pressure,  while  the  greater  part  of  the 
force  that  is  applied  to  them  is  borne  by  their  mutual  action  up- 
on each  other's  surfaces.  Arches,  generally  speaking,  have 
curved  or  polygonal  surfaces,  forming  the  lower  part  of  their 
mass,  which  are  concave  towards  the  horizon.  They  are  support- 
ed at  the  extremities  upon  walls  ;  and  in  this  case,  the  resistance 
they  oppose  to  the  forces  that  tend  to  destroy  them,  is  principally 
that  with  which  their  materials  resist  a  force  that  tends  to  crush 
them.  But  there  are  also  cases  in  which  the  arch  is  a  curve  or 
polygon,  convex  towards  the  horizon,  in  which  case  their  princi- 
pal resistance  is  due  to  the  absolute  strength  of  the  material. 

As  both  the  absolute  strength,  and  the  resistance  to  compres- 
sion are  more  intense  in  all  materials,  than  their  respective 
strength  ;  and  as,  in  addition,  the  forces  that  tend  to  destroy  an 
arch,  act  in  most  cases  obliquely,  it  is  at  once  obvious,  that  an 
aperture  of  far  greater  length  can  be  covered  by  an  arch,  than 
can  be  done  by  any  other  application  of  the  same  material. 

The  circumstances  that  affect  arches  will  differ  according  to 
the  materials  of  which  they  are  constructed.  These  are  princi- 
pally, stone  or  brick,  cast  iron,  wood,  wrought  iron  in  the  shape 
of  chains,  and  ropes  ;  arches  of.  the  three  former  substances  re- 
quire no  distinctive  appellation,  we  shall  call  arches  of  the  last 
two  kinds,  Arches  of  Suspension. 

207.  A  stone  or  brick  arch  is  composed  of  a  number  of  prisms, 
whose  section  is  a  trapezium  ;  these  may  be  considered  as  trun- 
cated wedges,  and  are  called  Voussoirs.  They  are  generally  of 
an  uneven  number  ;  the  odd  one,  which  occupies  the  vertex  of 
the  arch,  is  called  the  Keystone.  The  vertical  walls  on  which 
they  rest,  are  called  Abutments,  and  when  there  are  two  con- 
tiguous arches,  the  intermediate  wall  is  called  a  Pier.  The  point 
where  the  vertical  wall  meets  the  curve  of  the  arch,  is  called  the 
Spring  of  the  arch.  The  distance  between  the  piers  or  abutments 
that  support  a  single  arch,  its  Span.  The  lower  or  inner  curve 


EQUILIBRIUM  OP  F Book  ///. 

of  an  arch  is  called  its  Intrados;  the  upper  or  outer  curve,  its 
Extrados.  The  intervals  between  the  voussoirs  are  called 
Joints. 

208.  In  considering  the  theory  of  arches,  it  has  been  usual  to 
proceed,  either  by  assuming  a  given  intrados,  and  investigating 
the  relative  size  of  the  voussoirs  ;  or  assuming  the  magnitude  of 
the  voussoirs,  to  investigate  the  curve  of  the  intrados.  In  both 
cases,  the  voussoirs  have  been  considered  as  wedges,  perfectly  free 
to  move  upon  each  other,  or  resisted  neither  by  friction  nor  the 
tenacity  of  cement. 

By  the  former  method,  it  may  be  shown,  that  the  magnitude 
of  the  voussoirs,  in  order  that  equilibrium  should  exist,  should 
be  to  each  other  as  the  portions  cut  from  a  horizontal  line  passing 
through  the  vertex  of  the  arch,  by  the  prolongations  of  the  joints. 
In  the  case  of  an  arch,  whose  intrados  is  a  portion  of  a  circle,  the 
relative  weights  of  the  voussoirs  would  be  the  differences  of  the 
tangents;  and  if  the  arch  were  semicircular,  the  lower  voussoirs 
must  be  infinite. 

By  the  latter  method,  it  may  be  shown,  that  the  arch  of  equili- 
brntion,  if  the  voussoirs  be  equal,  must  be  a  homogeneous  cate- 
naria  ;  and,  in  the  case  of  unequal  voussoirs,  a  catenarian  curve 
loaded  with  weights  proportioned  to  those  of  the  voussoirs.  And 
in  the  case  of  weights,  varying  in  the  relation  of  the  differences  of 
the  series  of  tangents,  the  curve  would  become  a  portion  of  a  circle. 

Both  methods  being  founded  upon  the  same  hypothesis,  give, 
not  only  in  the  last  quoted  instance,  but  in  all  others,  similar  re- 
sults. 

An  arch  of  equilibration,  determined  in  either  mode,  would 
have  its  centre  of  gravity  in  the  highest  possible  position,  and 
would  therefore  be  in  a  state  of  tottering  equilibrium.  Hence,  any 
action,  however  small,  exerted  upon  it,  would  shake  the  voussoirs 
from  their  position,  and  the  arch  would  be  destroyed.  So  far  is  this 
from  being  the  case  in  practice,  that  arches  may  be  considered 
among  the  most  permanent  of  structures  ;  hence,  it  is  obvious, 
that  so  far  from  the  friction  and  the  adhesion  of  the  cements 
being  quantities  that  may  be  safely  neglected,  or  for  which  a 
mere  correction  may  be  applied  1o  an  hypothesis  from  which 
they  are  first  abstracted,  they  constitute,  in  truth,  forces  as  essen- 
tial to  the  conditions  of  equilibrium,  as  the  mutual  pressure  of  the 
voussoirs  themselves. 

-  it  would  also  appear  from  these  hypotheses,  that  an  arch  would 
exert  no  horizontal  thrust  upon  its  abutments,  unless  the  face  of 
the  abutment  were  not  a  tangent  to  the  arch  at  its  spring;  and 
hence,  that  a  thrust  would  not  be  created  by  flattening  the  arch, 
provided  that  the  radius  of  curvature  at  the  spring  coincided  with 


ARTIFICIAL  8TRUCTUKES. 


201 


EookllL] 

the  horizontal  line  passing  through  it.     All  these  inferences  are, 
in  like  manner,  contradicted  by  experiment. 

In  order,  then,  to  the  establishment  of  a  true  theory  of  arches, 
it  is  necessary  that  experiment  and  observation  should  be  pre- 
viously called  in,  to  show  the  exact  circumstances  under  which 
arches  change  their  form,  or  actually  give  way.  Such  experi- 
ments were  first  made  on  very  small  models,  by  Davisy,  at  JVlont- 
pelier,  about  the  year  1732  ;  and  were  repeated  on  a  larger  scale 
by  Boistard,  in  1800.  Similar  experiments  were  also  made  by 
Cauthey  ;  and  the  same  author  has  recorded  his  observations  up- 
on a  number  of  broken  bridges,  and  others  in  such  a  state  of  de- 
cay as  to  require  their  being  taken  down.  Perronet  likewise,  at  a 
previous  date,  had  made  accurate  observations  upon  the  change 
of  figure  undergone  by  new  arches,  at  the  moment  of  removing 
their  centres. 

209.  Arches  are  built  by  laying  their  voussoirs  upon  a  tem- 
porary arch  or  frame  of  wood,  called  a  Centre,  whose  upper  sur- 
face has  the  form  it  is  intended  to  give  to  the  arch.  In  laying 
the  voussoirs,  it  is  found  that  the  lower  ones  retain  their  position 
simply  by  virtue  of  the  friction  upon  the  faces  on  which  they 
rest,  and  that  they  may  be  laid  wholly  independently  of  the 
centre.  They  do  not  begin  to  slide,  until  the  inclination  of  the 
faces  becomes  equal  to  about  40°.  At  this  time  they  begin  to 
press  upon  the  centre,  which  would  have  its  form  changed  in  con- 
sequence, were  it  not  loaded  at  the  summit,  for  the  purpose  of 
counteracting  this  change  of  figure. 

As  the  number  of  voussoirs  increases,  a  pressure  begins  to  take 
place  on  the  upper  part  of  the  centre,  which  tends  to  press  it 
against  the  spring  of  the  arch  ;  the  upper  voussoirs  tend  to  turn 
around  their  lower  angle,  and  the  joints  open  at  the  extrados. 
When  the  keystone  is  placed,  and  the  centre  removed,  the  open 
joints  close  again,  and  new  openings  and  motions  appear  in  the 
arch,  to  represent  which,  we  must  have  recourse  to  a  figure: 


The  upper  parts  of  the  arch,  from  D  to  d,  are  no  longer  sup- 
ported, except  by  their  mutual  pressure  ;  this  tends  to  close  their 

26 


202  EQUILIBRIUM  OV  [Book  III. 

joints  at  the  extrados,  and  press  them  from  each  side  towards  the 
point  E  :  this  point  becomes  a  point  of  support  for  both  halves  of 
the  arch.  The  joints  of  the  upper  part  of  the  intrados  will  open. 
The  pressure  on  the  point  E,  necessarily  reacts  towards  the  abut- 
ments and  the  lower  parts  of  the  arch,  which  it  tends  to  over- 
throw by  causing  them  to  turn  around  their  outer  angles  K  and  Ar. 
In  consequence  of  this  pressure  and  reaction,  each  half  of  the  arch 
separates  into  parts  at  some  intermediate  points,  D  or  d.  These 
serve  as  points  of  support  for  the  higher  parts,  and  to  transmit 
their  action  towards  the  abutments.  If  the  latter  do  not  possess 
sufficient  stability  to  resist  the  pressure  of  the  arch,  it  separates 
into  four  parts,  which,  in  breaking,  turn  around  the  points  K,D,E, 
d  and  AJ,  as  upon  hinges.  If  the  abutments  are  capable  of  meet- 
ing the  pressure,  it  still  manifests  itself  by  closing  the  joints  of 
the  extrados  near  the  point  E,  and  of  the  intrados  near  the  points 
d  and  D  ;  causing  those  of  the  intrados  to  open  near  the  point  E, 
and  of  the  extrados  near  the  points  D  and  d. 

The  position  of  the  points  d  and  D,  which  are  called  Points  of 
Rupture,  depends  upon  the  figure  of  the  vault,  and  the  distribu- 
tion of  the  weight  it  supports.  In  determining  the  strength  of 
an  arch,  it  is  important  to  know  their  position.  It  is  always  at 
the  weakest  part  of  the  arch  ;  but  this  is  not  necessarily  that  where 
it  is  thinnest,  as  we  shall  see  hereafter. 

In  the  experiments  it  was  found,  that  : 

In  semicircular  arches,  that  did  not  rest  on  abutments,  the 
points  of  rupture  were  at  an  angle  of  30°  from  the  spring. 

In  oval  arches,  formed  of  three  circular  arcs,  at  50°,  of  the 
small  arc  rising  from  the  spring. 

In  flat  arches,  the  points  of  rupture  were  at  the  spring,  as  they 
were  in  all  circular  arches  whose  versed  sine  was  less  than  a 
fourth  part  of  the  chord. 

In  all  cases  the  whole  mass,  say  the  arch  and  its  abutments, 
tended  to  divide  into  four  parts,  turning  upon  points  in  the  intra- 
dos, at  the  spring  or  base  of  the  abutment,  and  the  vertex,  and 
separating  at  two  intervening  points. 

210.  To  investigate  the  action  of  the  forces  that  tend  to  produce 
these  motions. 

Suppose  the  arch  to  be  divided  into  two  symmetric  parts,  by 
the  vertical  plane  passing  through  the  line  EC,  the  relations  of 
the  forces  will  be  identical  on  each  side  of  EC.  Let  K  be  the 
origin  of  the  co-ordinates  ;  let  x  and  y  be  the  horizontal  and  ver- 
tical co-ordinates  of  the  point  D  ;  x'  and  y'  the  co-ordinates  of  the 
point  E. 

Resolve  the  forces  that  act  at  D  into  two,  Xand  Y,  parallel  to  the 
jtwo  co-ordinates  ;  those  that  act  at  E  into  two  also,  X'  and  Y'. 


///.]  ARTIFICIAL  STRUCTURES.  203 

By  the  principle  of  vertical  velocities,  §  70,  the  equation  of  equi- 
librium will  be 


'=0.  (218) 

The  forces  that  act  at  D,  act  upon  a  lever  KD,  which  we  shall 
caliyj  those  at  E  upon  another  DE,  which  we  shall  call  g-,  their 
values  in  terms  of  the  co-ordinates  are 


g=^  [(*'-*)"+  (»'-»)']; 

the  differentials  of  which  are  respectively 


and 

(a?'—  ar).  (dxf—  dx)  +  (y'—y)-(dy'—dy)  =0. 
Multiplying  the  first  members  of  these  equations  respectively  by 
the  constant  co-efficients  /  and  /',  we  have 

Ixdx  -\-lydy  ; 
and 

l'(x'—x')dxt—l(x>—x}dx+  &c. 

If  these  be  added  to  the  equation  (218),  we  shall  find  from  it 
that  the  sum  of  the  terms,  that  involve  the  differential  of  any  one 
of  the  co-ordinates  =0,  or 

TLdx+lxdx—  l'(x'—  x)dx=Q  ; 
T[dy+lydy—l'(y'—y]dy=0  ; 
r'—  x)dx'=0  ; 


dividing  by  dx,  dx',  dy  and  cfy',  we  obtain 


and  eliminating  /  and  /', 


')jr—  (Y+T>=0.  I  (219) 

The  forces  which  act  are  the  weights  of  the  two  portions  of  the 
arch,  applied  to  their  respective  centres  of  gravity,  M  and  N  ;  call 
these  weights  m  and  n. 

The  arch  being  in  equilibrio, 

X=0. 

If  for  m  we  substitute  its  two  parallel  components  acting  at  D  and 
E,  their  magnitudes  will  be 

FQ 

at  E,     m         ; 


204  EQUILIBRIUM  OF  [Book  III. 

EF 
and  at  D,     wi         =Y'. 


The  force  m  resolved  in  like  manner.,  gives  for  its  action,  at  D, 

KS 


hence  the  sum  of  the  forces  that  act  at  D,  is 
_    EF        KS 

Y~mEQ"'"nKR; 
and 

DQ 


FQ 

because  it  is  only  the  component  of  m  JTQ  in  the  direction  of  E 

that  acts.  Substituting  these  values  in  the  two  equations,  (219) 
we  obtain  identical  values  for  both  terms  of  the  first  member  of 
the  first  ;  in  the  second,  we  have 

S—  »»«KR=0.  (220) 

In  order,  then,  that  equilibrium  shall  exist,  the  first  term  must  be 
equal  to  the  sum  of  the  two  last  ;  and  for  stability  it  ought  to  be 
greater. 

The  deductions  from  this  theory  are  abundantly  simple. 

From  the  value  of  the  horizontal  thrust,  X',  it  appears  that  it 
increases  with  the  length  of  the  line  FQ,  or  with  the  approach  of 
the  centres  of  gravity  of  the  upper  parts  of  the  arch  to  their  sum- 
mit. 

We  may,  therefore,  see  that  it  would  be  possible  to  investigate 
a  curve  for  the  extrados,  when  the  intrados  is  given,  that  would 
form  an  arch  without  any  horizontal  thrust  whatsoever.  When 
the  centre  of  gravity  of  the  higher  part  of  the  half  arch  falls  in  the 
middle,  which  it  will  very  nearly  do,  when  the  arch  is  very  low 
and  the  vault  of  uniform  thickness, 


and  the  formula  becomes 

m    DO 

—  .  ^  .  KU—  n.KS—  «i.KR=0.  (221) 

If  the  arch  be  flat,  EQ  becomes  equal  to  the  thickness  of  the 
voussoirs.     If  we  call  this  thickness  e,  and  the  span  of  the  arch 

-  ,  the  expression  will  become 

m     / 

-«-  .  -  .  KU—  n.KS—  w.KR=0.  (222) 

£         6 


Book  III.]  ARTIFICIAL  STRUCTURES.  205 

From  this  it  follows,  that  the  horizontal  thrust  of  a  flat  arch,  is 
equal  to  the  fourth  part  of  its  weight,  multiplied  by  the  ratio  of 
the  length  to  the  thickness.  Not  only  does  the  horizontal  thrust 
diminish  with  the  approach  of  the  centre  of  gravity  of  the  upper 
part  of  the  half  arch  to  its  vertex,  but  the  same  causes  a  diminu- 
tion in  the  pressure  on  the  keystone. 

In  very  low  arches,  this  pressure  becomes  equal  to  the  eighth 
part  of  the  weight  of  the  whole  arch,  multiplied  by  the  relation  of 
the  span  to  the  whole  height  of  the  arch,  measured  from  the  top 
of  the  keystone. 

In  flat  arches,  the  pressure  on  the  keystone  is  just  half  the  hori- 
zontal thrust. 

211.  It  will,  therefore,  be  at  once  seen,  that  it  is  advantageous 
to  diminish  the  depth  of  the  keystone  as  much  as  possible  ;  for 
this  will  diminish  both  the  pressure  upon  itself,  and  the  horizon- 
tal thrust.  No  more  depth,  therefore,  should  be  given  to  the  key- 
stone, than  is  just  sufficient  to  insure  safety  from  the  joint  action 
of  the  pressure  upon  it,  and  the  agitation  produced  by  extrinsic 
causes.  The  value  of  the  resistance  to  pressure  in  various  kinds 
of  stone,  has  been  given  in  §191.  It  will  be  proper  to  allow  for 
its  diminution  in  the  ratio  of  the  squares  of  the  depth,  which  in 
this  case  will  be  the  horizontal  dimension  of  the  keystone,  and  to 
proportion  the  surface  so  as  to  bear  three  times  the  weight  that 
is  capable  of  crushing  it. 

In  arches  of  the  form  of  a  circular  arc  of  but  few  degrees,  the 
horizontal  thrust  is  very  great;  and  even  when  a  sufficient  thick- 
ness is  given  to  the  abutment,  absolute  safety  is  not  obtained  :  for 
the  effort  of  the  thrust  may  be  sufficient  to  overcome  the  tenacity 
of  the  cement,  and  separate  the  haunches  of  the  arch  from  the  abut- 
ment. This  risk  is  even  greater  in  arches  of  wood  or  iron,  rest- 
ing upon  stone  abutments,  and  there  are  several  instances  on 
record,  of  bridges  of  these  materials  having  fallen  as  soon  as  the 
centering  was  removed. 

The  first  application  of  the  formula  (220),  is  to  the  determina- 
tion of  the  position  of  the  points  of  rupture.  When  the  figure  of 
the  arch  is  given,  the  resolution  of  this  problem  presents  no  great 
difficulty  in  its  principles.  These  are  as  follow :  An  arch  ne- 
cessarily breaks  in  its  weakest  part,  which  is  that  in  which  its  re- 
sistance has  the  least  ratio  to  the  forces  that  act  upon  it,  and  which 
is  not  in  consequence'  necessarily  the  thinnest  part  of  the  vault. 
The  point  of  rupture  will  therefore  be,  where  the  moment  of  the 
force  that  tends  to  overturn  the  lower  part,  is  the  greatest  possi- 
ble, when  compared  with  the  forces  that  tend  to  sustain  it. 


206  EQUILIBRIUM   OF  [Book  III. 

This  ratio  is 

FQ  DQ 


roKR-f-nKS  ;  (223) 

when  this  is  a  maximum,  its  differential=0. 

In  circular  arches,  however,  this  simple  principle  is  attended 
with  practical  difficulties,  in  consequence  of  the  transcendental 
quantities  which  the  nature  of  the  circle  introduces.  In  arches 
formed  of  several  circular  arcs,  the  calculation  becomes  wholly 
impossible.  In  place,  then,  of  a  direct  mathematical  investigation, 
the  position  of  the  points  of  rupture,  determined  by  the  experi- 
ments of  which  we  have  spoken,  is  assumed  as  the  basis  of  the 
calculation.  The  resulting  thicknesses  of  the  abutments,  is  con- 
siderably less  than  is  usually  given  in  practice,  and  requires  to 
be  increased,  in  order  to  allow  for  want  of  firmness  in  the  founda- 
tion ;  and  to  augment  the  pressure  on  the  plane,  where  the  arch 
meets  the  foundation,  so  much  as  by  the  friction  to  prevent  sli- 
ding, should  the  adhesion  of  the  mortar  be  insufficient.  A  pow- 
erful aid  might  be  obtained  in  resisting  the  latter  action,  by  uni- 
ting the  courses  of  stone  by  means  of  dowels.  We  give  the  thick- 
ness of  abutments,  calculated  by  Gauthey,  whose  work  we  have 
followed  in  our  analysis,  for  arches  of  60  French  metres  in  span; 
this  will  serve  as  a  guide  in  similar  inquiries. 


Species  of  Arch. 

Thickness  of  Abut- 
ments. 

Position  of  Points  of 
Rupture. 

METKES. 

Semicircular,                      1.  32 
Oval  flattened  id,               1.  62 
do         do     ':  th              2.  24 
Circular  arc  of  60°,            3.09 

13°  30' 

31°  30' 
40«  so' 
0°  00' 

212.  It  has,  generally  speaking,  been  usual  to  give  to  the  piers  of 
bridges  the  same  thickness  as  to  their  abutments.  But  this  is  by 
no  means  necessary  ;  for  when  two  arches  rest  on  the  same  pier, 
their  horizontal  thrusts  mutually  counterbalance  each  other  ;  and 
the  pier  has  no  other  stress  to  bear  than  that  of  the  superincum- 
bent weight.  It  is  frequently  advantageous  in  practice  to  make 
the  piers  as  thin  as  possible  ;  as  for  instance,  when  a  bridge  cros- 
ses a  rapid  stream,  or  one  subject  to  sudden  floods.  The  ancient 
bridges,  and  those  of  England,  generally  speaking,  have  their 
abutments  equal  to  their  piers;  and  this  is  absolutely  necessary 
when  the  arches  are  built  in  succession.  But  when  all  the  arches 
are  carried  up  simultaneously,  and  their  centres  struck  at  the  same 
time,  it  becomes  practicable  to  give  dimensions  to  the  piers  suited 
merely  to  the  stress  they  are  afterwards  to  sustain.  Such  is  the 
present  practice  of  the  French  engineers. 


Book  III.}  ARTIFICIAL  STRUCTURES.  207 

213.  Arches  belong  as  a  principal  feature  to  no  architecture 
more  early  than  that  of  the  Romans.  Belzoni  indeed  states,  that 
he  found  at  Thebes,  the  relics  of  arches  of  brick  that  seemed  to- 
be  of  a  date  prior  to  the  Persian  conquest.  But  even  were  the 
arch  then  known,  it  was  but  little  used.  The  oldest  arch  in  ex- 
istence is  that  of  the  Cloaca  Maxima  at  Rome,  the  architects  of 
which  were  Etrurians;  and  in  the  works  of  the  Romans,  we  find 
arches  superior  in  magnitude  to  any  that  have  hitherto  been  con- 
structed, with  but  one  exception,  that  we  shall  presently  state. 
The  arches  of  the  bridge  of  Trajan,  over  the  Danube,  were 
semicircular,  raised  on  lofty  piers  and  abutments,  and  had  a  span 
of  ISO  feet. 

The  bridge  of  Vieillebrioude,  in  France, approaches  more  nearly 
to  the  bridge  of  Trajan  in  dimensions,  than  any  other  modern 
bridge;  it  was  built  in  1454,  and  has  a  span  of  178  feet.  And  as 
the  bridge  of  Trajan  has  long  since  fallen,  it  was  until  within  a 
year,  the  largest  arch  in  existence. 

The  bridges  of  Gignac  and  Lavaur,  in  France,  have  each  arches 
of  160  feet. 

*    The  arches  of  the  bridge  of  Neuilly,  and  the  centre  arch  of  the 
bridge  of  Mantes,  have  each  127  feet  span. 

The  great  arch  of  the  bridge  of  Verona,  in  Italy,  has  160  feet. 
The  marble  bridge  at  Florence,  built  by  Michael  Angelo,  has 
138  feet. 

The  span  of  the  centre  arch  of  Waterloo  bridge,  in  London,  is 
120  feet,  and  of  that  of  the  new  London  bridge,  140  feet. 

At  the  present  moment  a  bridge  is  constructing  over  the  Dee, 
at  Chester,  in  England,  whose  span  is  200  feet.  Should  this  stand 
after  the  centre  is  removed,  it  will  be  the  greatest  stone  arch  of 
ancient  or  modern  times. 

In  consequence  of  the  abundance  and  excellence  of  other  mate- 
rials in  the  United  States,  our  stone  bridges  are  neither  numerous 
nor  important.  The  most  beautiful  specimen  of  this  species  of 
architecture  that  we  possess,  is  the  acqueduct  bridge  at  the  Little 
Falls  of  the  Mohawk. 

Equilibrium  of  Domes. 

214.  The  same  principles  are  applicable  to  domes  or  spherical 
vaults  ;  and  if  the  curve,  in  the  figure  on  p.  201,  that  represents 
a  section  of  an  arch,  be  now  assumed  for  the  section  of  a  dome,  the 
circumstances  that  take  place  in  its  fracture  will  also  be  represented, 
it  having  points  ot  fracture,  and  dividing  into  four  parts.  But  in 
arches  which  are  cylindrical,  or  vaults  that  are  cylindroidal,  the 
rupture  takes  place  in  the  horizontal  plane  passing  through  the 


203  EQUILIBRIUM  or  [Hook  III. 

points  D  and  dt  while  in  a  dome,  the  line  of  rupture  is  a  circle.  In 
domes,  the  weight  of  the  upper  portions  will  be  less  than  in  arches, 
the  points  of  rupture  will  lie  higher,  the  horizontal  thrust,  and  the 
pressure  on  the  keystone  will  also  be  less.  Domes,  therefore,  will 
be  borne  by  less  abutments  than  arches ;  but  as  domes  are,  gene- 
rally speaking,  raised  upon  a  lofty  wall,  instead  of  resting  upon  an 
abutment,  it  is  inconvenient  to  give  this  wall  a  great  thickness; 
and  hence  artificial  means,  that  will  be  presently  mentioned,  are 
resorted  to,  in  order  to  give  the  requisite  resistance. 

215.  Domes  are  of  more  easy  construction  than  arches,  for  each 
course  keys   itself,   and   they  may  even  be   erected  without  any 
permanent  centre  ;   while  an  arch  must  be  finished,  and  the  key- 
stone placed,  before  it  supports  itself;  each  course  in  a  dome  serves 
as  a  keystone  to  those  that  are  beneath,  and  apertures  of  large  di- 
mensions may  therefore  be  left  in  the  middle  of  domes.     These 
serve  for  the  admission  of  light,  or  may  be  made  the  base  of  other 
more  elevated  parts  of  the  structure. 

Domes  derive  much  of  their  beauty  from  a  geometrical  property 
they  possess,  which  is  as  follows:  the  common  intersection  ofa« 
sphere,  and  a  cylinder  whose  axis  is  directed  to  the  centre  of  the 
sphere,  is  a  circle,  and  of  course  a  plane  curve.  Hence,  if  any  num- 
ber of  arches  be  arranged  on  the  sides  of  a  regular  polygon,  a  dome 
may  be  built  resting  upon  them  ;  and  a  cylindrical  tower  may  be 
built  upon  the  opening  in  the  centre  of  a  dome,  and  may  in  its 
turn  become  the  support  of  a  second  dome. 

216.  Domes  had  their  origin  among  the  Etruscans,  whose  tem- 
ples were  circular  in  plan,  and  covered  with  a  simple  hemispheric 
vault.     The  Romans  borrowed  this  species  of  structure  from  that 
neighbouringnation,and  brought  it  to  great  perfection.  The  finest 
antique  specimen  that  remains  of  this  species  of  building,  is  that 
which  goes  by  the  name  of  the  Pantheon.       This  building  has  a 
circular  ground  plan,  on  which  is  raised  a  lofty  cylindric  wall  that 
bears  a  hemispheric  dome,  144  feel  in  diameter.   As  the  walls  have 
not  of  themselves  sufficient  stability  to  support  with  certainty  the 
lateral  thrust,  they  are  loaded  at  ihe  spring  of  the  arch  by  ma- 
sonry, accumulated  in  the  following  mode:  The  generating  curve 
of  the  inner  surface,  or  intrados  of  the  dome,  is  a  semicircle,  the 


Book  IIL]  ARTIFICIAL  STRUCTURES.  209 

extrados  is  a  less  portion  of  a  greater  circle,  as  represented  in  the 


n 


B 

figure  ;  hence  it  becomes  possible  to  raise  the  external  vertical 
faces  AB,  of  the  wall,  much  higher  than  the  internal  CD;  and  a 
great  weight  rests  upon  the  spring  of  the  arch,  while  the  lower 
portions  of  the  arch  itself  are  also  strengthened. 

The  domes  of  a  number  of  Christian  churches  were  built  at 
the  intersection  of  the  aisles  that  form  a  cross  ;  they  were  hence 
borne  upon  four  arches,  to  which  they  were  applied  upon  the 
principle  stated  in  the  last  section. 

To  give  greater  elevation,  the  first  dome  was,  in  the  progress 
of  architecture,  terminated  immediately  above  the  keystones  of 
the  arches,  and  a  second  dome  raised  upon  the  circular  ring  that 
constituted  the  opening.  A  bolder  architect  proposed  to  raise  a 
cylindric  wall  upon  this  ring,  and  support  upon  it  the  second 
dome.  Thus  was  gradually  reached  the  sublime  conception  of 
the  dome  of  St.  Peter's.  In  realizing  this  conception,  various 
practical  difficulties  presented  themselves.  The  principle  ap- 
plied in  the  Pantheon  was  inapplicable,  in  consequence  of  the 
great  mass  of  material  it  would  require,  that  might  have  increased 
the  pressure  beyond  that  which  the  abutments  of  the  supporting 
arches  could  bear.  A  flattened  external  dome  would  have  been 
invisible,  except  from  a  distance,  and  wholly  deficient  in  beauty. 
For  these  reasons,  the  dome  that  was  seen  from  within,  was  made 
of  the  smallest  practicable,  but  of  uniform  thickness,  and  the  part 
seen  from  without,  the  half  of  an  oblong  spheroid.  In  the  dome  of 
St.  Peter's,  these  two  domes  are  both  of  masonry,  and  spring  from 
a  common  base,  as  in  the  figure,  diverging  as  soon  as  the  outer 

and  innner  curve  of  each,  intersect 
each  other.  In  the  case  of  St.  Paul's, 
in  London,  the  inner  dome  is  of  brick, 
the  outer  a  wooden  frame,  bearing  a  co- 
vering of  sheet  lead.  To  support  the 
frame  that  bears  the  outer  dome,  a  trun- 
cated cone  of  brick  rises  from  the  inner 
dome,  and  bears  the  smaller  cupola  or 

27 


210  EQUILIBRIUM    0?  \BuokUL 

lantern,  which  in  St.  Peter's  is  borne 
on  the  ring  that  terminates  the  outer 
dome.  A  section  of  the  dome  of  St. 
Paul's  is  represented  in  the  annexed 
figure. 

Although  in  these  different  modes, 
lightness  and  beauty  were  both 
gained,  the  resistance  to  the  hori- 
zontal thrust  is  so  much  diminished, 
that  the  domes  could  not  have  been  supported  by  the  balancing 
of  their  parts,  or  by  the  cohesion  of  cement;  to  remedy  this  de- 
fect, the  lower  courses  of  the  dome  of  St.  Peter's  are  bound  by 
strong  hoops  of  wrought  iron  ;  and  at  St.  Paul's,  chains  are  laid 
in  a  groove,  cut  in  the  stone  ring  whence  the  dome  springs,  and 
secured  by  melted  lead. 

Domes  and  groined  arches  are  formed  in  Gothic  architecture, 
at  the  junction  of  vaults  formed  by  the  intersection  of  two 
circular  arcs.  The  lateral  thrust  is,  in  these  cases,  met  by  but- 
tresses formed  on  the  principles  of  §198,  and  the  resistance  to  it 
is  further  increased,  by  loading  the  buttresses  with  heavy  masses 
of  stone,  assuming  the  form  of  pinnacles.  These,  which  form 
one  of  the  chief  embellishments  of  Gothic  architecture,  are  beau- 
tiful, not  only  from  their  graceful  figures,  but  from  their  evident 
adaptation  to  an  important  purpose. 

Of  Wooden  Arches. 

217.  Wood  may  be  applied  to  the  construction  of  arches  also, 
or  to  the  formation  of  frames  that  may  answer  as  a  substitute  for 
arches.     The  application  of  the  principles  employed  in  the  pre- 
ceding chapter,  to  this  material,  is,  however,  different  from  that 
which  is  adapted  to  the  theory  of  stone  arches.     While  in  stone 
arches  the  mass  is  made  up  of  separate  parts,  which  divide  in  par- 
ticular points,  and  move  around  others  as  if  they  had  no  cohesion, 
so  that  the  respective  strength  may  be  considered  as  nothing, 
and  the  whole  available  resistances  are,  that  of  friction,  and  that 
which  the  material  opposes  to  a  crushing  force  ;  the  use  of  wood 
brings  into  efficient  action  its  respective  strength,  and  the  resist- 
ance to  separation  may,  in  some  cases,  become  that  furnished  by 
the  absolute  strength.     The  length  that  can  be  safely  given  to  a 
single  beam,  supported  or  fixed  at  each  end,  and  lying  in  a  hori- 
zontal position,  is  limited  by  its  own  weight,  as  has  been  seen  in 
§  186. 

218.  If  the  force  that  acts  be  greater  than  can  be  borne  by  a 
single  beam,  two  may  be  united  in  such  a  manner  as  to  act  like 


Book  Hlt]  ARTIFICIAL  STRUCTURES.  211 

a  single  piece.     Thus,  if  a  beam  be  merely  laid  upon  another,  as 


T> 


AB  upon  CD,  each  will  resist  the  effort  to  bend  it,  which  must 
precede  fracture,  merely  with  its  own  force;  the  system  is  there- 
fore only  twice  as  strong, in  respect  to  the  sum  of  the  forces  that  act, 
as  the  single  beam.  If  then,  thesingle  beam  have  reached  thelimitat 
which  it  breaks  by  its  own  weight,  they  will  both  be  broken. 
If  now  the  two  beams  be  united,  as  may  be  simply  done  by  dow- 
els or  pins,  in  such  a  way  that  one  cannot  bend  without,  causing 
an  equal  bending  in  the  other,  the  two  will  act  as  a  single  beam, 
and  the  strength  will  be  four  times  as  great  (141),  as  that  of  the 
single  beam,  or  twice  as  great  as  the  united  strength  of  the  two 
acting  separately  ;  hence,  the  two  thus  united,  will  now  not  only 
bear  their  own  weight,  but  require  an  additional  weight  equal  to 
their  own,  to  break  them.  Two  beams  may  be  still  more  ad- 
vantageously united  by  cutting  their  adjacent  surfaces  into  the 
form  of  the  teeth  of  a  saw,  turned  in  opposite  directions,  as  in  the 
figure.  If  these  be  united  by  screw  bolts,  or  .by  straps  of  metal, 


both  must  bend  together,  and  hence  act  to  resist  fracture  like  a 
single  beam.     This  method  is  called  Trussing. 

219.  When  the  limit  of  strength  is  reached,  either  in  a  single 
beam  or  in  this  arrangement,  two  beams  may  next  be  placed  in 
an  inclined  position,  pressing  against  each  other,  as  in  the  figure. 


The  action  of  the  weight  being  now  oblique  to  the  direction  of 
the  beam,  the  effort  of  the  weight  will  decrease  with  the  cosine 
of  the  angle  of  inclination,  §  184. 

Let  F  be  the  strength  of  the  compound  beam,  and  W  the 
weight  just  sufficient  to  break  it. 

F=Wcos.z. 

A  lateral  thrust  will  also  take  place  at  the  points  A  and  B,  which 
may  be  represented  by 

Wcot. ».  (224) 


212  EQUILIBRIUM  07  [Book  III 

220.  If  instead  of  two  beams  inclined  at  an  angle,  a  piece  of 


-A.  B 

small  curvature  be  substituted,  the  effort  of  the  weight  will  be 
diminished,  §  134,  in  the  ratio  of  the  cosine  of  the  angle  CAB  j 
and  the  resistance  of  the  wood  to  flexure,  as  well  as  its  strength, 
will  be  enhanced  even  in  a  higher  ratio,  in  consequence  of  the 
difficulty  of  bending  a  curved  piece  in  the  direction  opposed  to 
its  curvature.  The  latter  advantage  is  of  course  gained  only  in 
the  case  of  the  fibres  of  the  wood  being  also  curved  ;  for,  if  the 
fibres  be  cut  across,  the  strength  will  be  diminished  ;  because  the 
lateral  cohesion  of  the  wood,  which  (see  §  180)  is  far  less  than 
its  respective  strength,  is  now  the  only  resistance  that  remains. 

Whenever  the  line  CA  does  not  cut  the  intrados,  we  may  with- 
out error  consider  the  half  of  the  arch  as  a  straight  piece  of 
equal  dimensions,  loaded  at  one  end  by  a  weight  acting  vertically, 
and  standing  itself  upright.  This  weight  will  be  equal  to  so 
much  of  the  force  as  acts  in  the  direction  AC.,  and  the  condition 
of  equilibrium,  between  this  component  of  the  whole  of  the  dis- 
turbing forces  that  act,  and  the  strength  of  the  arc  will  be  given 
by  the  formula  (199). 


in  which  I  is  the  length  of  the  arc  CA. 

We  have  in  the  preceding  chapter  omitted  the  question  of  the 
elasticity  of  wood,  and  may,  in  this  case,  substitute  for  that  pro- 
perty the  respective  strength.  Using  this,  it  will  be  obvious  that 
there  are  three  different  methods  of  valuing  E,  according  as  the 
pieces  that  form  an  arch,  when  there  are  more  than  one,  are  com- 
bined :  when  they  all  act  distinctly  ;  when  they  are  merely  com- 
bined by  the  vertical  posts  of  the  arch,  or  by  other  pieces  crossing 
them  ;  and  when  they  are  so  united  as  to  form  a  body,  no  part  of 
which  can  move  without  affecting  all  the  rest. 

In  the  first  case,  supposing  each  of  the  pieces  to  have  a  rectan- 
gular section,  whose  breadth  is  o,  and  depth  6  ;  let  10  be  the  num- 
ber from  the  table  of  relative  strength,  n  the  number  of  pieces. 

E=na62M>;  (225) 

In  the  last  case, 

E=n2a62tp;  (226) 

and  in  the  second  case  it  will  be  intermediate. 

It  is,  however,  hardly  possible  to  unite  beams  in  such  a  way  as 


Book  III.]  ARTIFICIAL  STRUCTURES.  213 

to  give  them  the  entire  strength  determined  by  the  last  formula. 
And  in  any  case  whatever,  a  large  allowance  ought  to  be  made 
after  calculating  the  value  of  E. 

If  W  be  a  weight  resting  upon  the  vertex  of  the  arch, 

Q=W  cos.  i. 

But  if  the  weight  be  uniformly  distributed  over  the  arch, 
Q=iW  cos.  i. 

221.  If,  however,  the  curvature  be  considerable,  we  can  no 
longer  consider  the  wooden  arc  as  formed  of  two  straight  pieces, 
but  must  have  recourse  to  principles  in  some  respects  similar  to 
those  of  stone  arches.  In  the  arc  ACB,  a  force  acting  at  C,  will 

not  break  the  beam  at  that  point, 
but  at  two  points,  D  and  d,  inter- 
mediate between  it  and  the  two 
points  of  support;  and  the  lower 
parts,  AD  and  Bc?,will  turn  around 
A  and  B,  as  in  the  case  of  a  stone 
arch,  (see  §  209).  The  point  of 
rupture  is  easily  determined  in 
this  case  ;  for,  the  respective 
strength  of  a  -beam  of  equal  thick- 
ness, being  uniform  throughout,  if  the  momentum  of  the  stress 
be  abstracted,  whether  the  beam  be  straight  or  crooked,  the  rup- 
ture must  occur  where  the  effortof  the  weight  acts  most  directly 
upon  the  beam.  This  point  will  be  determined  by  letting  fall  a 
perpendicular  from  the  centre  of  the  arc,  c,  upon  the  line  CA. 

Having  thus  determined  the  point  of  rupture,  we  may  proceed 
to  determine  the  conditions  of  equilibrium  between  the  force  P 
that  acts  in  the  direction  CA,  and  the  resistance  at  the  point  D. 
This  may  be  done  by  conceiving  that  the  part  DA  is  immovea- 
ble,  and  that  the  part  DC  being  firmly  fixed  at  D,  is  acted 
upon,  by  a  force  that  tends  to  bend  it,  in  the  direction  C  A.  This 
will  not  affect  the  condition  of  equilibrium,  and  it  becomes  an 
application  of  the  formula,  (192). 


,   .. 

In  which  /  is  the  length  of  the  arc  CA,  and  /the  versed  sine  of 
the  curvature  which  the  force  P  is  capable  of  producing.     From 
this  we  obtain  for  the  value  of  P 
3E/ 

'"~T'  (227) 

But  /  is  a  function  of  P,  and  the  value  of  the  latter  is  still  in- 
volved in  the  expression,  it  is,  therefore,  necessary  to  have  the 
means  of  determining  /.  This  can  always  be  safely  done  by 


EQUILIBRIUM  OF  [Book  II  L 

taking  as  the  value  of/,  the  maximum  flexure,  or  that  which  pre- 
cedes rupture.     This  deduced  from  experiment,  is 


222.  In  applying  these  principles  to  the  construction  of  bridges, 
two  different  methods  have  been  pursued. 

(1.)  A  continuous  arc  has  been  formed  by  bending  plank,  ar- 
ranged so  that  none  of  their  joints  should  be  opposite,  and  united 
by  bolts  and  iron  straps,  in  such  a  manner  as  to  act  as  a  single 
piece.  Such  is  the  principle  of  the  bridges  erected  in  Europe 
under  the  direction  of  Wiebeking.  Of  these  the  bridge  of  Bam- 
berg  is  the  most  remarkable.  Its  span  is  221  feet. 

In  this  bridge,  the  road  passes  over  the  summit  of  the  arch, 
which  is  therefore  flat,  and  has  a  great  lateral  thrust.  If  this  be 
not  carefully  opposed  by  a  proper  connexion  with  the  abutments, 
and  by  giving  them  a  sufficient  weight,  the  bridge  may  be  de- 
stroyed by  it.  This  was  the  case  in  two  bridges  erected  on  a 
similar  principle  at  Paterson,  New-Jersey. 

It  is,  however,  by  no  means  necessary,  except  where  it  is  de- 
sired to  leave  room  for  the  passage  of  vessels,  to  make  the  road 
pass  over  the  vertex  of  the  arch.  The  nature  of  the  material, 
which  is  both  light  and  strong,  admits  of  the  arch  being  formed 
of  several  separate  ribs.  Carriages  and  passengers  may  pass  be- 
tween these  upon  a  horizontal  road,  resting  on'  timbers,  support- 
ed by-thearch  from  above.  If  the  timbers,  supported  by  each 
rib,  be  made  to  form  a  continuous  chord,  and  be  connected  with 
the  rib  at  the  spring  of  the  arch,  they  will,  in  addition,  oppose 
their  absolute  strength  to  the  horizontal  thrust,  which  may  thus 
be  entirely  done  away.  The  ribs,  too,  may  be  made  with  a  much 
greater  curvature,  and  will  be  both  stronger  in  consequence,  and 
have  less  horizontal  thrust. 

Such  is  the  principle  of  the  very  beautiful  bridge  erected  by 
Burr,  over  the  Delaware  at  Trenton,  N.  J.,  the  larger  arches  of 
which  have  194  feet  span. 

(2.)  An  arch  of  timber  may  be  formed  of  pieces  arranged  in  the 
form  of  a  polygon;  such  an  arch  would  be  in  equilibrio,  had  it 
the  form  of  the  funicular  polygon,  §  29  ;  but  as  the  equilibrium 
would  be  tottering,  it  is  better  to  make  the  system  rigid,  in  which 
case,  it  is  unnecessary  to  seek  or  observe  the  law  of  equilibrium. 
The  system  rnay  be  made  rigid  by  extending  some  of  the  timbers 
beyond  the  points  where  they  intersect  the  others,  framing  them 
together,  and  connecting  them  again  with  others,  forming  trian- 
gular frames,  which  cannot  alter  their  shape  without  breaking. 
Such  is  the  principle  of  the  wooden  bridges  of  Hampton,  and 
Cambridge,  in  England.  The  largest  of  these,  however,  has  less 


///.]  ARTIFICIAL  STRUCTURES.  215 

than  50  feel  span.  This  method  is  extremely  faulty,  and  no 
bridge  of  any  great  span  constructed  after  it,  has  been  of  long  du- 
ration. A  better  mode  of  making  the  system  rigid,  is,  to  make 
it  double,  and  interpose,  between  the  arch  pieces,  queen-posts, 
AB,  A'B'?  &e.  forming  by  means  of  mortices  and  tenants,  a  series 


of  open  voussoirs,  or  quadrilateral  frames.  To  prevent  these 
from  changing  their  figure,  a  frame  of  the  figure  of  a  St.  Andrew's 
cross  is  placed  in  each.  This  principle  was  adopted  in  the  bridge 
erected  over  the  Piscataqua,  near  Portsmouth,  New-Hampshire, 
whose  span  is  256  feet. 

(3.)  The  two  methods  may  be  combined;  of  this  we  have  an 
instance  in  the  great  arches  constructed  by  Wernwag,  one  over 
the  Schuylkill,  near  Philadelphia,  the  other  over  the  Delaware 
at  Easton.  The  former  has  340  feet  span,  and  is  the  largest 
wooden  arch  now  in  existence.  A  project  for  an  arch  upon  this 
principle,  of  400  feet  span,  is  given  by  Tredgold,  in  his  work  on 
the  principles  of  carpentry. 

223.  In  all  the  methods  of  extending  wooden  structure  across 
openings,  of  which  we  have  hitherto  spoken,  the  principle  of  the 
arch  has  been  taken  as  the  leading  and  prominent  feature.  Far 
more  simple  considerations  have  led  to  the  construction  of  wooden 
bridges,  of  greater  span  than  any  we  have  hitherto  cited. 

The  simplest  mode  of  spanning  an  opening  is  a  beam  ;  but  we 
have  seen  that  this  has  an  early  limit  in  the  size  of  timber,  which 
cannot  be  obtained  of  sufficient  depth  to  enable  a  long  beam  to 
bear  its  own  weight.  It  might,  however,  occur,  that  as  it  is  pos- 
sible by  trussing,  as  explained  in  §  218,  to  obtain  beams  of  great- 
er strength  than  single  trees  will  afford,  so  it  would  be  practicable 
to  build  a  structure  which  should  act  upon  the  principle  of  a  beam. 
The  first  thing  that  ought  to  be  determined,  for  this  purpose,  is 
the  figure  that  would  have,  under  equal  size,  the  greatest  degree 
of  strength;  such  is  one  that  would  have  the  moment  of  its 
strength  equal  in  every  part  of  its  length. 

The  beam  having  a  rectangular  section,  whose  constant  breadth 
is  a,  and  variable  depth  v,  the  strength  of  any  section,  if  supported 
at  both  ends,  will  be  (152) 


EQUILIBRIUM    OF  [Book  III. 


the  resistance  to  flexure  will  be  (203) 
E 


P 

If  we  call  the  force  that  acts  to  bend  it,  P,  we  shall  have  for 
the  condition  of  equilibrium 

Px     E 

-=-=  —  2ot?3.  (228) 

2        p 

If  the  weight  be  uniformly  distributed,  we  shall  have  for  the 
value  of  P,  in  terms  of  the  weight  w,  borne  by  the  unit  of  the 
beams'  length, 


and  the  equation  (228)  becomes 
E 


whence 

x      ow 

r¥<S«  (229) 

which  is  the  equation  of  a  straight  line,  making,  with  the  horizon, 

pw 
an  angle,  whose  tangent  is   -/(p—  )•   And  as  the  circumstances 

are  similar  on  each  side  of  the  middle  of  the  beam,  ^the  solid 
of  greatest  strength  is  an  isosceles  triangle.  We  cannot,  how- 
ever, diminish  the  thickness  of  the  beam  at  its  points  of  support 
to  0,  and  hence  the  figure  becomes  a  pentagon,  as  represented 
beneath,  two  of  whose  sides  are  parallel  and  equal,  and  two  of 
whose  angles  are  right  angles. 


If  we  examine  the  action  of  the  weight"  to^cause  this  beam  to 
bend,  we  shall  see  that  the  fibres  nearest  the  upper  part  would 
be  compressed,  and  those  nearest  the  lower  would  be  lengthened, 
and  thus  a  line  acb  might  be  drawn,  which  would  separate  the 
extended  from  the  contracted  fibres.  Such  a  line  is  called  the 
Neutral  Axis.  In  arranging  the  pieces  of  wood  of  which  the 
bridge  is  composed,  the  best  method  will  obviously  be,  that  which 
shall  bring  their  absolute  strength  most  nearly  into  direct  oppo- 
sition, to  the  contractions  and  expansions,  which  the  beam,  if  of 
a  single  piece,  would  undergo  in  bending,  but  shall  which  give  them 


Book  ///.] 


ARTIFICIAL    STRUCTURES. 


217 


a  firm  bearing  on  the  abutments.  These  pieces  may  then  be 
united  by  vertical  posts,  which  will  form  the  whole  into  a  series 
of  open  triangular  frame  work ;  to  these  posts  may  be  attached 
a  horizontal  trussed  beam,  which  will  answer  to  bear  the  road. 
Such  a  plan  of  frame  work  is  represented  beneath,  and  it  will 
be  obvious,  that  the  horizontal  beam  will  destroy  the  lateral 
thrust. 


This  is  the  principle  that  was  adopted  by  a  Swiss  carpenter  of 
the  name  of  Grubenman,  in  the  construction  of  the  bridges  of 
Schaff hausen  and  Wettingen.  The  first  of  these  had  a  span  of 
365  feet,  and  appeared  to  be  divided  into  two  spans,  resting  on 
an  intermediate  pier.  But  the  use  of  this  support  was  in 
opposition  to  the  desires  of  the  architect,  and  he  had  the  skill  to 
leave  it  questionable  whether  the  bridge  derived  any  strength 
from  it  or  not. 

The  bridge  of  Wettingen  had  a  span  of  384  feet.  Both  of 
these  remarkable  bridges  were  of  sufficient  strength  to  bear  any 
probable  load.  Both  were  unfortunately  destroyed  during  the 
campaign  of  1799,  and  neither  have  been  replaced. 

When  from  the  position  of  the  bridge,  the  road  must  pass  over 
its  summit,  the  beam  beneath  may  be  suppressed,  and  the  system 
takes  the  form  represented  in  the  figure,  which  is  that  of  the 


bridge  of  Kandel,  in  the  canton  of  Berne,  constructed  by  Rit- 
ter.  In  this  there  is  a  horizontal  thrust,  which  must  be  counter- 
acted by  the  resistance  of  the  piers. 

A  modification  of  the  same  form,  proposed  by  Gauthey,  is  re- 
presented beneath. 


It  may  be  objected  to  the  bridges  of  Grubenman,  that  on  many 

28 


218 


EQUILIJ1KIUAI    OF 


[Book  III. 


parts  of  them  the  moments  of  the  forces  that  tend  to  change  the 
figure  become  very  greatj  in  consequence  of  the  great  length  of 
the  arms  on  which  they  act.  On  this  account,  a  frame  of  this 
description  has,  in  many  cases,  been  combined  with  a  bent  arc 
of  plank,  like  that  of  the  bridges  of  Wiebeking  and  Burr. 
When  this  is  introduced,  the  resistance  it  opposes  to  bending 
seems  sufficient,  and  no  other  stress  need  be  guarded  against  ex- 
cept that  which  tends  to  absolute  fracture.  In  some  of  the 
bridges  proposed  by  Gauthey,  therefore,  the  long  pieces  that  are 
extended  to  prevent  the  former  action,  are  suppressed,  and  the 
arch  takes  the  following  form: — 


A  still  better  arrangement,  founded  on  the  same  principle,  is 
to  be  found  in  the  bridge  of  Wernwag,  that  crosses  the  Delaware, 
at  New  Hope.  The  principle  of  this  is  exhibited  beneath. 


M 


_•• 


An  ingenious  application  of  the  principle  of  the  beam,  has 
been  made  by  an  American  engi-neer,  ElhicI  Town.  His  bridge 
is  composed  of  a  rectangular  frame  of  timber,  connected  by  di- 
agonal braces. 

This  bridge  will  have  great  strength  to  resist  fracture,  but  little 
to  resist  flexure,  in  consequence  of  the  length  of  the  horizontal 
beams,  and  the  mobility  at  the  angles  of  the  braces.  It  is,  how- 
ever, lighter  than  any  other  plan  that  has  ever  been  proposed, 
and  might,  by  a  few  obvious  improvements,  be  made  capable  of 
spanning  great  openings. 

Of  Cast  Iron  Arches. 

224.  Cast  iron  bridges  may  be  erected  either  of  continuous 
ribs  and  bands,  or  by  forming  the  material  into  skeleton  vous- 
soirs.  In  the  first  case,  their  theory  is  the  same  as  that  of  wood- 
en arches,  substituting  the  resistance  of  cast  iron  for  that  of  wood  ; 
in  the  second  case,  their  theory  is  similar  to  that  of  stone  arches, 


ARTIFICIAL  STRUCTURES.  219 

substituting  the  value  of  the  strength  of  the  one  material  for  the 
other.  Stone,  a  material  of  small  respective  strength,  and  more 
than  double  the  weight  of  wood,  requires  that  the  intrados  should 
have  a  continuous  surface,  and  the  space  between  it  and  the  ex- 
trados  is  often  of  necessity  filled  up.  Wood,  a  material  of  great- 
er respective  strength,  of  less  resistance  to  a  crushing  force,  and 
less  weight,  may  be  arranged  in  separate  and  similar  ribs,  and  a 
lightness  in  the  arch,  far  more  than  proportioned  to  the  different 
densities  of  the  substances,  being  thus  attained,  spans  of  far  great- 
er extent  may  be  compassed  than  by  stone.  Cast  iron  holds  an 
intermediate  rank  between  these  two  materials  :  being  possessed 
of  more  strength  than  either,  it  has  three  times  the  density  of 
stone;  but  it  may,  like  wood,  even  if  formed  into  voussoirs,  bear- 
ranged  in  separate  ribs ;  the  voussoirs,  too,  need  no  more  ma- 
terial than  will  form  a  flaunch  around  their  periphery,  and  furnish 
supporting  braces  within.  Still,  iron  will  form  an  arch  much 
heavier  than  wood  ;  and  the  limit  at  which  it  is  crushed  by  its 
own  weight  is,  therefore,  sooner  reached. 

Of  the  two  methods  which  have  been  mentioned,  that  of  con- 
tinuous ribs  is  objectionable,  because  cast  iron,  froru  its  crystalline 
structure,  has  no  great  tenacity  ;  hence  voussoirs  are  better, 
by  means  of  which  the  resistance  to  crushing  is  substituted  for 
the  respective  strength. 

225.  After  the  full  manner  in  which  the  theories  of  stone  and 
wooden  arches  have  been  discussed,  it  is  unnecessary  to  enter  in- 
to a  repetition  of  the  principles,  even  for  the  purpose  of  applying 
them  to  another  material.      We  shall,  therefore,  be  content  with 
giving  a  list  of  the  principal  arches  of  cast  iron  that  have  hither- 
to been  constructed. 

The  first  instance  of  an  iron  bridge  is  that  of  Coalbrookdale 
in  England.  Its  span  is  100  feet.  The  main  rib  is  composed  of 
no  more  than  two  pieces,  meeting  at  the  vertex  of  the  arch.  It 
is,  therefore,  an  instance  of  the  first  kind  of  cast  iron  arch.  It 
was  erected  about  the  year  1776.  The  second  cast  iron  bridge 
is  at  Build  was,  near  Coalbrookdale.  Its  span  is  131  feet.  The 
date  of  its  erection  is  1795. 

One  at  Sunderland,  in  England,  over  the  Wear,  erected  in 
1796,  has  a  span  of  209  feet.  It  is  the  first  instance  of  the  for- 
mation of  cast  iron  into  voussoirs. 

The  bridge  of  Austerlitz,  at  Paris,  has  five  arches  of  106  feet 
each,  and  is  composed  of  skeleton  voussoirs,  very  scientifically 
arranged. 

Of  Chain  Bridges. 

226.  Bridges  may  be  supported  by  means  of  chains,  or  ropes. 


220  EQUILIBRIUM  OP  [Book  HI. 

In  the  earlier  forms,  planks  were  laid  directly  upon  the  the  flexi- 
ble material,  which  was  stretched  between  two  firm  supports  ;  the 
chain  or  rope  was,  for  the  convenience  of  passage,  brought  into  a 
position  as  nearly  horizontal  as  possible.  This  method  is,  however, 
inconvenient  in  practice,  and  diminishes  the  resistance  of  the  ma- 
terial ;  for  it  will  be  seen  by  reference  to  §27,  that,  considering 
the  bridge  as  a  funicular  polygon  of  an  infinite  number  of  sides, 
the  effort  of  a  weight  acting  in  a  vertical  direction  to  break  it,  will 
be  much  increased  by  diminishing  the  curvature  ;  and,  after  all, 
the  road  could  never  cease  to  have  an  inconvenient  slope. 

A  far  better  application  of  the  principle  was  carried  into  effect 
about  the  year  1796,  by  Mr.  James  Findley  of  Pennsylvania.  In- 
stead of  attempting  to  render  the  road  nearly  level  by  the  tension 
of  the  chains,  he  made  them  of  such  length  that  the  versed  sine 
should  be  not  less  than  one  seventh  of  the  span.  The  road  was 
suspended  from  the  chain,  and  might  therefore  be  rendered  per- 
fectly horizontal.  Chains,  iron  rods,  or  beams  of  wood  may  be 
used  to  effect  the  suspension.  Forty  bridges  of  this  description 
were  erected  iu  the  United  States  previous  to  the  year  1808.  As 
wrought  iron  is  the  material  that  has  the  greatest  absolute 
strength;  as  the  chains  by  which  the  road  is  suspended  may  be 
multiplied,  and  the  longitudinal  beams  having  thus  many  points  of 
support,  need  not  be  very  thick;  and  as  wood  is  the  lightest  ma- 
terial of  which  a  road  can  be  constructed,  it  is  susceptible  of  de- 
monstration, that  an  arch  of  greater  span  may  be  constructed  upon 
the  principle  of  Mr.  Findley,  than  in  any  of  the  modes  that  we 
have  yet  spoken  of,  or  indeed  in  any  other  manner  that  has  yet 
been  proposed.  It  is  only  wonderful  that  chain  bridges  have  not 
yet  come  into  more  general  use,  for  there  are  innumerable  cases 
in  which  they  possess  greater  advantages  than  those  of  any  other 
material. 

It  was  not  until  1814,  that  Findley's  method  attracted 
the  attention  of  European  engineers.  At  that  date  it  was 
in  contemplation  to  shorten  the  post  road  from  London  to  Liver- 
pool, by  effecting  a  passage  across  the  Mersey  at  Runcorn,  a 
position  in  which  no  other  material  would  have  been  applicable ; 
for  the  locality  required  an  arch  with  a  span  of  1000  feet.  Tel- 
ford,  therefore,  proposed  a  bridge  composed  of  a  timber  road 
suspended  by  chains,  identical  in  principle  and  form  with  those 
of  Findley;  and  having  instituted  a  series  of  interesting  and  useful 
experiments  on  the  strength  of  wrought  iron,  he  proved  beyond 
all  question  the  practicability  of  the  project.  It  has  not  however 
been  carried  into  effect. 

In  1S18,  a  bridge  formed  of  iron  wire,  bearing  a  path  for  foot 
passengers,  which  had  been  erected  the  year  before  by  the  Earl 
of  Buchan,  across  the  Tweed  at  Dry  burgh,  was  carried  away, 


///.]  ARTIFICIAL  STRUCTURES.  221 

and  replaced  immediately  by  a  chain  bridge,  which  was  the  first 
erected  in  Europe  on.Findley's  principle.  About  the  same  period, 
Telford  presented  a  project  for  one  of  similar  character,  across 
the  straits  of  the  Menai,  which  has  since  been  executed. 

The  latter  is,  in  point  of  extent  of  span,  the  most  remarkable 
bridge  in  existence.  The  distance  between  the  centre  of  the  piers 
is  560  feet,  arid  the  road  is  elevated  100  feet  above  the  level  of  the 
highest  tides.  The  height  of  the  supports  above  the  level  of  the 
road,  which  height  corresponds  with  the  versed  sine  of  the  arc, 
was  intended  at  first  to  have  been  no  more  than  35  feet,  but  has 
been  increased  to  50  feet. 

Before  the  completion  of  the  bridge  over  the  Menai,  Captain 
Brown,  so  well  known  as  the  introducer  of  chain  cables  for  ships, 
completed  the  construction  of  a  chain  bridge  over  the  Tweed, 
near  Berwick  :  this  was  opened  to  the  public  in  1820,  and  was 
the  first  executed  in  Europe,  which  was  adapted  for  the  convey- 
ance of  loaded  carriages.  Since  that  period,  numerous  other 
bridges,  and  several  piers  for  the  reception  of  vessels  have  been 
constructed  in  Great  Britain,  and  the  method  has  been  successfully 
introduced  into  France. 

227.  A  bridge  formed  of  chains  with  a  road  suspended  from 
it,  is  in  the  condition  of  the  funicular  polygon,  §  29.  Its  theory 
may,  however,  be  more  easily  reduced  to  calculation  Tor  practi- 
cal purposes,  by  considering  it  as  a  catenaria,  and  the  inequality 
in  the  distribution  of  the  weight  is  too  small  to  cause  any  error 
in  practice,  from  assuming  it  to  be  loaded  at  every  point  with  an 
equal  weight. 

If  we  take  one  of  the  points  of  suspension  of  the  chain,  for  the 
origin  of  the  co-ordinates,  x  and  y  ;  and  if  a  be  the  angle  the 
curve  makes  with  its  tangent  at  that  point  ;  w  the  weight  borne 
by  each  unit  of  the  length  of  the  chain,  assumed  to  be  constant  ; 
and  T  the  tension  of  the  chain  at  the  point  of  suspension,  we  have 
by  the  theory  of  the  catenaria,  §  28,  for  the  equation  of  the  curve 

A  cos.  a  (  A  —  wy±  \/[(A  —  wy}2  —  A2  cos.2  a]  ) 

~^~        g*   \  ~  A(l—  sin.  a)  ~~;  I 

for  the  versed  sine/,  equal  to  the  height  of  the  point  of  suspension 
above  the  level  of  the  horizontal  tangent  of  the  curve, 


for  the  half  span,  =  \  s, 

A  cos.  a  i    cos.  a    \ 

i  s=  —         -  log.  (-  -  .  --  )  ,  (446 

w  \l  —  sin.  a  / 

for  half  the  length  of  chain,  =  *  /, 


EQUILIBRIUM  OP  [Book  111. 

A  sin.  a 

*  '=  —i-  ;  («<) 

and  for  the  relation  between  /and  /, 

,1  —  cos.  a 

f=±l  -  —    - 
sin.  a 

from  (44a)  and  (44c),  we  obtain  for  the  values  of  A, 

id  wf 

A  =  —  -  -  =  -  -  ^  --  .  (230) 

2  sin.  a      1  —  cos.  a 

The  direct  determination  of  the  value  of  the  angle  a,  from  the 
properties  of  the  catenaria,  is  not  easy  ;  but  when  the  versed  sine 
does  not  exceed  T^  th  of  the  span,  the  curve  does  not  differ  sensi- 
bly from  a  circle,  that  would  have  the  same  lines  for  its  tangents 
at  the  points  of  suspension  :  from  the  properties  of  the  circle,  the 
value  of  the  tangent  of  the  angle  a,  is  found  in  terms  of  its  radius 
R,  and  chord  d,  to  be  as  follows,  v:z  : 

d 

~ 


and  the  value  of  the  radius  may,  from  the  properties  of  the  same 
curve,  be  found  by  the  subsidiary  formulae, 


' 

These  formulae  and  principles,  will,  generally  speaking,  suffice 
for  the  calculation  of  any  of  the  cases  that  can  occur  in  practice. 
But  fora  more  full  discussion  of  the  theory  in  which  all  the  cir- 
cumstances are  taken  into  account,  we  refer  to  the  work  of 
Navier,  Rapport  et  Memoire  sur  les  Ponts  suspendus,  Paris, 
1823. 

228.  From  this  we  quote  the  following  practical  rules  that  are 
immediate  inferences  from  his  investigations.  In  increasing  the 
span  of  chain  bridges,  there  is  no  reason  to  fear  that  the  changes 
of  figure,  growing  out  of  the  action  of  passing  loads,  will  be  aug- 
mented. These  changes  may  even  he  rendered  less  sensible,  in 
bridges  of  Wide§pan,  by  making  the  versed  sine  of  the  curve  bear 
a  less  proportion  to  the  extpnt  of  span.  In  fact,  as  the  ratio  of 
the  versed  sine  to  the  span  diminishes,  the  curve  of  the  bridge 
will  approach  more  and  more  near  to  a  straight  line,  and  at  this 
limit  the  figure  of  the  chains  is  invariable,  whatever  be  the  man- 
ner in  which  the  weight  is  distributed,  provided  they  be,  as  the 
hypothesis  assumes,  incxtensible.  On  the  other  hand,  by  dimi- 
nishing the  versed  sine,  the  constant  tension,  that  the  weight  of 
the  road  exercises  on  the  chains,  is  increased  also,  as  well  as  the 
variable  tension  growing  out  of  accidental  loads  ;  and  both  of  these 
tensions  would  become  infinitely  great,  if  the  chain  were  stretched 
in  a  straight  line. 


Book  III.']  ARTIFICIAL  STRUCTURES.  223 

229.  In  calculating  the  dimensions  of  the  iron  chains  which 
compose  the  inverted  arch,  as  well  as  those  by  which  the  road  is 
suspended  from  them,    the  absolute  strength,  §  179,   should   be 
made  at  least  twice  as  great  as  the  greatest  probable  tension  ;  for 
an  iron  rod  will,  at  a  limit  of  strain,  a  little  greater  than  the 
half  of  its  absolute  strength  begin  to  stretch;  and  its  elasticity  will 
be  so  much  impaired  by  the  strain,  that  it  will  not  restore  itself 
to  its  original  dimensions. 

230.  Iron  is  subject  to  fracture  on  sudden  changes  cf  tempera- 
ture.  This  appears  to  arise  from  unequal  expansion  ;  the  outer  parts 
being  sooner  affected,  expand  or  contract  earlier  than  those  within. 
To  prevent  any  danger  from  this  cause,  heavy  loads  should  not 
be  permitted  to  pass,  for  some  hours  after  any  great  and  sudden 
change  in  the  temperature  of  the  air  shall  have  occurred.     As 
changes  sufficient  to  cause  danger  are  by  no  means  frequent,  such 
precaution  cannot  be  productive  of  any  important  inconvenience. 

231.  Bridges  formed  of  chains  are  liable  to  two  species  of  os- 
cillations :  the  one  vertical,  growing  out  of  the  passage  of  weights ; 
the  other  horizontal,  arising  from  the  action  of  the  wind,  or  other 
external  disturbing  forces.     In  respect  to  the' former,  they  are 
more  likely  to  produce  injury  in  small  than  in  large  arches,  for 
both  the  extent  and  velocity  of  the  vibrations  decrease  with  the 
increase  of  the  span  ;  the  extent  of  this  kind  of  oscillation  follows 
the  inverse  ratio  of  the  squares  of  the  chord  of  the  curve  ;  while 
the  velocity  decreases  with  the  length  itself. 

232.  Had  the  chains  no  weight  to  support  besides  their  own, 
they  would  be  readily  caused  to  oscillate  in  a  horizontal  direction, 
and  would  follow  in  their  vibrations  the  law  of  pendulums,  the 
time  of  performing  them  being  independent  of  the  intensity  of 
the  disturbing  force.      The  mathematical  investigation    shows, 
that  the  amplitude  of  these  oscillations  would  diminish  in  a  ratio 
more  rapid  than  that  in  which  the  length  of  the  chains  increases. 

The  chains,  however,  are  connected  with  the  timber  road  in 
such  a  manner,  that  they  cannot  oscillate  in  a  horizontal  direction 
without  causing  it  to  change  its  figure,  either  horizontally  .or 
vertically.  As  it  is  easy,  by  a  proper  system  of  framing,  to  ren- 
der the  road  almost  inflexible  in  a  horizontal  direction,  these  os- 
cillations can  be  at  most  but  small,  unless  the  disturbing  force  be- 
come of  sufficient  intensity  to  cause  the  rupture  of  the  wood 
work.  So  long,  then,  as  the  tendency  to  oscillate  is  but  small, 
it  may  be  performed  without  meeting  with  much  resistance  ;  but 
so  soon  as  it  begins  to  increase,  it  is  opposed  by  the  whole  ri- 
gidity and  weight  of  the  timber  road. 


224  EQUILIBRIUM  OF  [  Hook  III. 

233.  The  strength  of  wood,   when  drawn  in  the  direction  of 
its  fibres,  being  very  great,  amounting,  §  ISO,  to  about  one-eighth 
part  of  the  strength  of  wrought  iron,  the  former  material  may  be 
used  to  suspend  the  road,  from  the  principal  chains,  to  great  ad- 
vantage, particularly  as  regards  economy  of  construction. 

234.  The  expense  of  the  chains  is  a  minimum,  when  the  versed 
sine  of  the  catenaria  has  to  its  chord  the  relation  of  1  :  2\/2  ;  but 
this  is  a  proportion  that  is  rarely  admissible  in  practice. 

235.  The  establishment  of  chain  bridges  offers  a  great  variety 
of  different  cases,  that  may  comport  with  arrangements  more  or 
less  simple  in  their  structure,  thus : 

(1.)  The  bridge  may  cross  a  ravine  situated  in  a  passage  en- 
closed between  rocks,  that  are  high  enough  to  afford  firm  and 
fixed  points  for  fastening  the  chains  ;  in  such  a  case,  and  particu- 
larly when  the  breadth  of  the  gorge  exceeds  4  or  500  feet,  a  chain 
bridge  may  not  only  be  the  most  economical,  but  often  the  only 
practicable  method  of  passage. 

(2.)  The  space  to  be  traversed  by  the  arch,  may  offer  firm 
supports  for  the  chains,  at  a  sufficient  height  on  one  side  only. 
In  this  case  the  curve  may  advantageously  have  the  form  of  a 
half  catenaria,  being  attached  at  a  proper  height  to  the  loft}'  bank, 
and  having  for  its  tangent  at  the  opposite  bank,  a  horizontal  line. 

(3.)  When  both  banks  are  low,  the  chains  must  be  attached  to 
artificial  supports.  These  are  sometimes  masses  of  masonry  ;  at 
others,  frames  of  cast  iron;  and  if  they  are  not  themselves  suffi- 
ciently solid  to  sustain  the  tension,  they  must  be  reinforced  by 
chains  extending  from  them,  in  directions  opposite  to  that  in  which 
those  which  support  the  bridge  are  suspended,  and  which  must  be 
firmly  fastened  to  the  ground. 

(4.)  This  last  mode  may,  in  some  cases,  be  modified  to  advan- 
tage, by  advancing  the  piers,  or  artificial  supports,  into  the  space 
that  is  to  be  traversed.  In  this  case,  the  chains  that  strengthen 
the  piers  are  made  to  bear  a  portion  of  the  bridge,  and  the  whole 
may  assume  the  arrangement  figured  beneath,  of  one  whole  and 


two  half  arches.     This  was  the  plan  proposed  by  Telford,  for 
the  bridge  at  Runcorn. 

(5.)  A  single  pier  may,  in  some  cases,  be  erected  in  the  mid- 
dle of  the  space,  and  the  bridge  take  the  form  of  two  half  cate- 


Book  III.']  ARTIFICIAL  STRUCTURES.  225 

nariae,  as  represented  in  the  figure.     A  bridge  of  this  shape  was 


constructed  in  England,  under  the  direction  of  Brunei,  to  be 
erected  in  the  Island  of  Bourbon. 

When  the  extent  of  the  proposed  bridge  is  great,  various  com- 
binations of  whole  and  half  arches  may  be  formed,  according  to 
the  local  circumstances. 

The  theory  would  show  that,  while  the  height  of  the  points 
that  support  the  chain  may  be  increased  indefinitely,  there  is 
no  practical  limit  to  the  extent  of  the  span,  except  that  at  which 
the  chains  would,  if  suspended  vertically,  break  by  their  own 
weight;  for  additional  strength  may  be  gained  by  increasing  the 
proportionate  magnitude  of  the  versed  sine  of  the  curve.  This 
investigation  is  however  of  little  value  in  practice,  for  besides  the 
immense  expense  of  artificial  supports  of  great  height,  the  oscil- 
lations are  rendered,  as  has  been  seen,  more  intense  by  such  an 
increase.  For  this  last  reason,  the  proportion  originally  em- 
ployed by  Findley  of  |,  has  been  reduced  to  ^  in  most  instances. 

If  the  proportion  between  the  span  and  the  versed  sine  is  con- 
stant, the  length  of  the  former  has  a  theoretic  limit.  This  has 
been  calculated  by  Navier,  under  the  assumption  that  the  relation 
is  y'j,  and  found  to  be  2900  feet.  As  no  bridge  has  yet  been 
erected  of  a  span  as  great  as  600  feet,  we  are  probably  still  be- 
neath the  practical  limit.  It  cannot,  however,  be  advised  to  at- 
tempt the  construction  of  chain  bridges  much  exceeding  the  last 
mentioned  extent,  and  the  increase  of  the  span  will  probably  be 
made  by  gradual  steps,  as  has  been  the  case  in  other  instances. 

236.  Bridges  of  wire,  and  round  iron  of  different  sizes,  have 
also  been  used;  in  these  the  road  rests  directly  upon  the  wires. 
More  recently,  in  France,  Seguin  has  proposed  to  use  iron  in 
this  form,  and  to  suspend  the  road  from  it,  as  in  the  chain  bridges 
of  Findley.  He  has  urged  in  favour  of  his  proposition  several 
plausible  arguments,  among  which  are,  the  greater  strength  ob- 
tained by  an  equal  quantity  of  metal  in  this  form,  and  the  entire 
removal  of  the  risk  that  arises  in  chains  from  an  imperfect  weld- 
ing of  the  links. 

29 


BOOK  IV. 

OF    THE  MOTION  OF  SOLID   BODIES. 

CHAPTER  I. 

OP  FALLING  BODIES. 

237.  From  what  has  been  said  in  the  preceding  book,  it  will 
be  seen,  that,  when  a  force  whose  direction  passes  through  the 
centre  of  gravity,  acts  for  an  instant  of  time  upon  a  body,  and 
then  abandons  it  to  itself,  the  motion  is  exactly  such  as  it  would 
have  were  all  tfre  matter  of  the  body  collected  in  that  centre, 
and  the  force  were  to  act  there  with  an  intensity  equal  to  that 
which  it  actually  has. 

We,  in  truth,  know  of  no  such  forces  in  nature.  No  body  can 
instantaneously  acquire  the  velocity  due  to  the  force  impressed, 
but  must  pass  through  inferior  degrees  of  motion,  requiring  a  cer- 
tain time  for  the  purpose.  But  there  are  innumerable  cases, 
where  this  time  is  so  small  as  to  be  absolutely  inappreciable,  and 
we  may  therefore,  without  committing  any  error,  assume  that 
the  action  is  instantaneous. 

So,  also,  as  every  body  is  composed  of  a  number  of  separate  par- 
ticles, which  in  solids  are  united  by  the  attraction  of  aggregation, 
(a  force  that  however  intense,  does  not  render  the  system  abso- 
lutely rigid  or  invariable,)  a  greater  or  less  time  will  be  required 
to  convey  the  impulse  from  the  point  to  which  it  is  originally  ap- 
plied, and  to  distribute  it  throughout  the  system.  This  time  is, 
like  the  other,  so  small  as  to  be  inappreciable ;  still  there  are  many 
cases  where  we  see  sure  indications  of  the  motion  having  been 
communicated  by  degrees  ;  and  there  are  even  some,  where  we 
take  advantage  of  this  circumstance  in  our  practical  applications. 

The  centre  of  gravity  of  any  body  acted  upon  by  a  force,  the 
duration  of  whose  action  is  so  small  as  to  be  insensible,  and  whose 
direction  passes  through  that  point,  moves  uniformly  forwards 
in  the  straight  line  which  marks  the  direction  of  the  force.  If 
two  such  forces  act,  the  centre  of  gravity  moves  uniformly  in 


228  OF    FALLING    BODIES.  [Book  IV. 

the  diagonal  of  a  parallelogram,  whose  sides  are  parallel  to  the 
direction,  and  represent  in  magnitude  the  intensity  of  the  forces. 
And  thus  of  any  number  of  forces,  according  to  the  principles  of 
§42. 

238.  When  a  body  falls  near  the  surface  of  the  earth,  it  is  acted 
upon  by  accelerating  forces,  whose  directions  are  perpendicular 
to  that  surface.  Within  the  small  space  that  any  body,  whose 
motion  can  usually  become  the  object  of  investigation,  occupies, 
these  forces  may  be  considered  as  parallel  to  each  other.  Their 
resultant,  then,  will  pass  through  the  centre  of  gravity,  to  which 
we  may  therefore  conceive,  that  a  single  accelerating  force  is  ap- 
plied. 

The  attraction  of  gravitation  to  which  this  action  is  due,  varies 
(§  100)  at  different  points  of  the  earth's  surface,  and  decreases  as 
we  rise  from  the  surface,  (§  9G),  according  to  a  determinate  law. 
Both  of  these  circumstances  may  aho  be  neglected  without 
causing  any  sensible  error;  and  hence,  a  body  left  without  sup- 
port, near  the  earth's  surface,  may  be  considered  as  a  body  moving 
from  rest,  under  the  action  of  a  constant  accelerating  force.  It 
will  therefore  move  in  a  straight  line,  in  the  direction  of  the  force, 
or  perpendicular  to  the  surface  of  the  earth,  and  with  an  uni- 
formly accelerated  velocity.  All  the  formulae  of  §  49,  are  there- 
fore applicable  to  this  case,  provided  a  value  be  assigned  to  g> 
the  measure  of  the  accelerating  force. 

By  the  methods  described  in  Book  III.,  Chap.  I.,  and  others 
to  which  we  shall  hereafter  refer,  it  has  been  inferred,  that  a  bo- 
dy moves  from  rest  in  a  second  of  time,  under  the  action  of  the 
force  of  gravity,  through  a  distance  of  16rV  feet.  Hence  we  have 
for  the  value  of  g,  (60),  321  feet. 

In  most  of  the  calculations  in  which  the  formulae  of  §  49  are 
employed,  it  is  sufficient  to  take  the  approximation  of 

g=32  feet. 
Substituting  this  value  we  obtain 


V  (231) 

2s 

T'~^'    J 

239.  A  heavy  body  projected  upwards  from  the  surface  of  the 
earth,  will  be  retarded  by  a  constant  force,  which  will  finally 
destroy  the  upward  motion.  It  will  then  begin  to  fall ;  the  mo- 
tion upwards  w^ill  be  equably  retarded,  the  motion  downwards 
equably  accelerated. 


Book  IV.  OF  FALLING  BODIES.  229 


For  the  time  of  flight,  and  the  height  to  which  it  rises,  we  have, 
by  the  substitution  of  the  same  value  of  g,  from  (61  a)  and 
(61  6), 


Hence  the  body  will  rise  to  the  height  whence  it  must  have 
fallen,  in  order  to  acquire  the  initial  velocity  ;  and  the  times  of 
ascent  and  descent  will  be,  exactly  equal.  The  whole  time  of 

i} 
flight  will  be  —  . 

240.  When  the  motion  of  bodies  falling  near  the  surface  of  the 
earth  is  actually  observed,  it  is  found  to  differ  materially  from 
what  would  be  obtained  from  the  calculation  of  the  preceding 
formulae.  Thus  Dr.  Desaguliers  found  that  a  ball  of  lead  of  two 
inches  in  diameter,  took  41  seconds  to  fall  from  the  dome  of  St. 
Paul's,  in  London,  to  the  pavement,  a  distance  of  272  feet.  Now 
by  the  formula  (231), 

s=l6l2. 

It  ought  therefore,  in  the  same  time,  to  have  fallen  through 
324  feet.  In  a  body  of  less  diameter  and  similar  density,  and  in 
one  of  equal  diameter  and  less  density,  the  difference  between 
the  formula  and  the  experiment  would  have  been  still  greater. 
The  cause  of  this  difference  is  the  resistance  of  the  air,  a  retarding 
force  analogous  to  friction,  but  which  follows  a  different  law. 

In  a  subsequent  part  of  the  work,  we  shall  examine  the  nature 
and  character  of  this  resistance.  For  the  present,  it  is  sufficient 
to  state,  that  it  is  usually  assumed  to  increase  in  the  ratio  of  the 
square  of  the  velocity.  This  is  nearly  true  when  the  distance 
through  which  a  body  falls  does  not  exceed  a  few  hundred  feet: 
and  it  therefore  may  not  be  wholly  useless  to  investigate  the  mo- 
tion of  falling  bodies  upon  this  hypothesis. 

Taking  the  notation  of  §  237,  the  resistance  of  the  air  may  be 
expressed  upon  the  above  hypothesis,  by  multiplying  v3  by  a  con- 
stant co-efficient.*  Call  this  co-efficient  g-Ar2,  the  retarding  force 
will  be, 


the  accelerating  force  of  gravity  being  g,  the  actual  force,/,  which 
accelerates  the  body's  motion,  will  be 

f=g-  ffW; 

/=g.(l—&V). 

From  (53)  and  (54),  we  may  deduce  the  equations 
/<&—  dr,  and  fds=vdv, 

*  Venturoli,  vol.  I.  p.  89 


230  OF  FALLING  BODIES.  [Book  IV. 

which  become 

dv          1 

^'=I=r  (233) 

f<fo 

zd*=l—kv-  J 

Integrating,  and  remarking  for  the  determination  of  the  constant 
quantity,  that  when  /=0,  we  have  at  the  same  time  s=0  and 
«?=0,  we  obtain  the  two  equations, 

I 


and  eliminating  r,  obtain  a  third, 

1  /  gkt       —gkt 

~ 


From  equation  (234)  it  will  be  seen,  that  the  greatest  possible 
value  of  v  cannot  exceed  T  .  Hence,  if  the  body  fall  from  a 

considerable  height,  the  velocity  may  finally  become  uniform. 

The  motion  of  a  rising  body  might  be  investigated,  by  taking 
for  the  retarding  force/, 

/=*+«*•• 

The  constant  co-efficient,  g-fc2,  is  -found  to  be  proportional  to 
the  density  of.  the  air,  and  in  bodies  of  similar  figures,  to  be  in  the 
inverse  ratio  of  their  homologous  dimensions,  and  of  the  density  of 
the  body.  Call  the  density  of  the  air  D',  let  m  be  a  constant  co- 
efficient, D  the  density  of  the  body,  and  r  the  homologous  di- 
mension, which,  in  a  spherical  body  is  the  radius,  we  have 


whence  we  have  for  the  value  of  k 


and  for  the  constant  velocity  which  cannot  be  exceeded, 


(flT  D 
£B> 


All  things  else  being  equal,  the  maximum  velocity  will  be  pro- 
portioned to  the  square  root  of  the  density  of  the  falling  body  ;  and 
hence,  the  denser  the  body,  the  longer  it  will  continue  to  be  acce- 
lerated, the  greater  will  be  the  constant  velocity  acquired,  and  the 
shorter  the  time  of  its  descent  through  a  given  distance. 


Book  IF.  OF    FALLING    BODIES.  231 


So  also,  all  things  else  being  equal,  the  constant  velocity  will 
be  proportioned  to  the  square  root  of  the  radii  of  spherical  bodies, 
and  hence  the  larger  the  body  of  the  same  material,  the  greater 
will  be  its  constant  final  velocity,  and  the  less  the  time  of  its  fall 
through  a  given  distance. 

241.  The  law  thus  ascertained  in  respect  to  bodies  falling 
through  the  air,  namely,  that  their  acquired  velocity  can  never 
exceed  a  certain  limit,  and  finally  becomes  uniform,  is  true  in 
all  eases,  where  a  body  impelled  by  an  accelerating  force  is  re- 
tarded by  another  force,  whose  intensity  increases  in  a  higher 
ratio  than  the  simple  velocity.  It  is  also  true,  as  will  be  obvious, 
when  the  accelerating  force  decreases  with  the  increase  of  the 
velocity,  and  the  retarding  force  is  constant. 

It  will  be  seen  from  (237)  that  if  the  density  of  the  air  vary, 
as  is  actually  the  case,  the  resistance  must  vary  also  ;  and  that 
the  co-efficient  of  the  square  of  the  velocity  cannot  be  constant, 
as  we  have  assumed  in  our  hypothesis,  but  will  be  less  in  rare 
than  in  dense  air. 

To  investigate  the  motion  of  a  falling  body,  in  such  a  manner 
as  to  include  all  the  circumstances,  it  would  be  necessary  then  to 
take  into  account  its  figure  and  density  ;  the  variation  in  the  in- 
tensity of  gravity  at  different  distances  from  the  surface  of  the 
earth  ;  the  variation  in  the  density  of  the  atmosphere  under  dif- 
ferent pressures  and  at  different  temperatures.  The  problem 
would  therefore  become  extremely  complex,  even  were  the  re- 
sistance of  air  of  a  given  density  to  increase  exactly,  as  assumed 
in  our  hypothesis,  with  the  square  of  the  velocity.  It  fortu- 
nately happens,  however,  that  there  are  few  or  no  cases  in  prac- 
tical mechanics,  in  which  a  greater  degree  of  accuracy  in  the  de- 
termination of  the  motion  of  falling  bodies  is  required,  than  is  to 
be  obtained  from  the  original  formulae  (231):  and  hence,  even 
the  investigation  we  have  copied  from  Venturoli,  and  which  may 
be  found  under  another  form  in  Poisson,  is  almost  wholly  a  mat- 
ter of  mere  curiosity. 

When  a  body  falls  from  rest  under  the  action  of  the  attraction 
of  gravitation,  as  the  direction  of  the  force  passes  through  its 
centre  of  gravity,  it  acquires  no  rotary  motion.  This  is  not  ne- 
cessarily the  case  when  it  is  projected  upwards,  for  the  projectile 
force  may  be  applied  to  a  point  other  than  its  centre  of  gravity  ; 
the  body  will,  in  consequence,  assume  a  rotary  motion  as  well  as 
one  in  a  vertical  direction.  It  therefore  becomes  necessary  that 
we  should  consider  the  circumstances  of  motions  of  this  character. 


232  ROTARY    MOTION  [Book  1 V. 


CHAPTER  II. 

OF  THE  ROTARY  MOTION  OF  BODIES. 

242.  It  has  been  shown,  §  82,  that  the  measure  of  the  moving 
force  of  a  body  is  the  product  of  its  mass  into  its  velocity.  This 
is  also  called  its  quantity  of  motion,  and  is  the  measure  of  a  force, 
that,  if  acting  for  an  instant,  and  then  abandoning  the  body  to 
itself,  would  communicate  to  it  the  given  velocity. 

A  solid  body  may  be  considered  as  a  number  of  material  points 
or  particles  of  matter  connected  with  each  other  in  such  a  way 
as  to  form  an  invariable  system.  If  each  of  these  points  be  acted 
upon  by  an  equal  and  parallel  force,  or  if  a  single  force,  or  the 
resultant  of  several,  be  applied  to  the  centre  of  gravity,  or  in  a 
line  whose  direction  passes  through  the  centre  of  gravity,  the 
body  will  move  in  a  straight  line  ;  and  as  the  points  will  be  in 
equilibrio  around  the  centre  of  gravity,  they  will  each  proceed 
also  in  a  straight  lino.  But  if  the  forces  that  act  upon  each  point 
be  not  equal  and  parallel,  or  the  resultant  do  not  pass  through 
the  centre  of  gravity,  each  point  will  have  a  tendency  to  niove 
in  the  direction  and  with  the  intensity  of  the  force  impressed. 
This  tendency  will  be  modified  by  the  mutual  connexion  of  the 
points  ;  and  hence,  although  each  forms  a  part  of  a  mass  whose 
general  velocity,  if  the  forces  cease  to  act,  is  constant,  yet  as 
each  will  have  a  different  velocity,  a  rotary  motion  must  ensue, 
around  some  point  comprised  within  the  system.  It  therefore 
becomes  necessary  as  a  preliminary  to  the  general  investigation 
of  the  conditions  of  the  motion  of  solid  bodies,  to  determine  the 
laws  of  rotary  motion. 

Let  us  first  assume  that  but  a  single  force  acts,  and  that  the 
body  must  revolve  around  a  fixed  axis. 

Let  ABCD  be  a  system  of  points  lying  in  one  plane,  and  in- 
variably connected  with  each  other,  and  with  an  axis  passing 
through  the  point  S,  whence  the  distances  to  the  points  respect- 
ively are  a,  6,  c ,  d ;  let  A  be  the  point  to  which  the  force  is 


Book  /F.J  o»  BODIES.  233 

applied,  and  v  the  velocity  it  would  have 
were  it  not  connected  with  the  other 
points,  u  the  velocity  it  has  in  conse- 
quence of  its  forming  a  part  of  the  sys- 
tem. The  quantity  of  motion  lost  by 
A,  on  this  account,  will  be 

A  (v-u)  ; 

and  the  moment  of  rotation  of  this  force, 
in  respect  to  S, 


As  the  whole  system  is  invariably  connected,  the  time  of  revo- 
lution of  all  the  points  will  be  the  same,  and  their  respective  ve- 
locities will  be  to  that  of  A,  as  their  distances  from  S  ;  hence 
their  respective  velocities  will  be 

bu  cu  du 

a  a  a 

The  quantities  of  motion  acquired  by  B,  C,  &  D,  will  there- 
fore be 

B  bu  C  cu  Ddu 


and  their  respective  moments  of  rotation 


u  Cc'u  Dd2^ 

T~  '  a      '          T~  ' 

By  the  principle  of  D'Alembert,  these  several  moments  of  ro- 
tation must  be  in  equilibrio  with  each  other,  or 

.      B6'W+ 
Aa  (t?  —  w)— 

whence 

A  a,  v 


The  moment  of  the  system  in  respect  to  the  point  S,  will  be 

(A  a2+B  62+C  c'-r-D  d2)  ~;  (239) 

each  point  in  the  system  will  therefore  exert  a  force  determined 
by  multiplying  it  by  the  square  of  its  distance  from  the  axis,  and 
by  the  velocity  at  A. 

The  sum  of  these  products,  extended  to  any  number  of  points, 
Aa2+B62+CcH-DdM-&e.  (240) 

is  called  the  Moment  of  Inertia  of  the  system,  in  respect  to  the 
fixed  point. 

The  angular  velocity  is  -,  which  is  the  value  of  the  angle  de- 
30 


234  ROTARY  MOTION  [Book  IV. 

scribed  around  the  point  S,  in  a  second  of  time,  in  parts  of  the 
radius  ;  and  to  determine  it  in  portions  of  a  circle,  the  quantity 


-  must  be  multiplied  by  57°.  29578  ;  the  value  of  the  arc  that  is 

equal  to  the  radius. 

For  the  value  of  the  angular  velocity  we  have  from  (238) 


These  propositions  are  obviously  true  of  a  system  composed 
of  any  number  of  points  whatever,  situated  in  the  same  place. 
They  are  also  true  of  a  system  lying  in  different  planes,  the  dis- 
tances a,  6,  c,  &c.,  being  in  this  case  the  perpendiculars  let  fall 
from  the  several  points  upon  the  fixed  axis. 

243.  In  any  system  of  points,  or  bodies,  that  are  compelled  to 
revolve  around  a  fixed  axis,  there  may  be  found  a  point  in  which 
if  they  were  all  collected,  a  given  force  applied  at  any  distance 
from  the  axis  will  communicate  the  same  angular  velocity  as  if 
it  were  applied  at  the  same  distance  from  the  axis  to  the  system, 
in  its  original  state.  This  point  is  called  the  Centre  of  Gyration. 

To  find  the  centre  of  gyration,  in  the  same  system  that  we 
have  just  considered. 

Let  x  be  the  distance  of  the  centre  of  gyration  from  the  axis  : 
the  moment  of  inertia  of  the  system,  if  united  in  this  point,  will  be 
(A-f-B+C+D)*2; 

u 
and  as  the  angular  velocity  -  is  to  be  the  same  in  both  cases,  this 

must  be  equal  to  the  moment  of  inertia  of  the  system  in  the  ori- 
ginal state,  or 

(A+B-fC+D)  ^=Aa2-r-B6a-f  Cc2+Dda, 
whence 


»=  ^    "      A+B+C+D  •  (242) 

In  order  to  apply  this  to  the  case  of  a  solid  body,  the  number  of 
points  must  be  supposed  infinite  :  call  each  of  them  dm,  and  the 
variable  distance  from  the  axis,  r,  the  formula  will  become 


(f9*d 
JdZ 


244.  There  will  be  a  point  in  the  radius  SA,  to  which  if  an 
obstacle  be  applied  sufficient  to  stop  the  rotary  motion  of  the 
system,  there  will  be  no  motion  communicated  to  the  axis  S. 
And  if  a  resistance  be  there  applied,  the  whole  of  the  force  of  the 
system  will  be  exerted  to  overcome  it.  This  point  is  called  the 
Centre  of  Percussion.  In  this  point  then,  if  all  the  matter  in  the 


Book  IV '.1  OP  BODIES.  235 


system  were  collected,  the  moment  of  rotation  will  be  equal  to 
that  of  the  system  in  its  original  state. 

To  find  the  position  of  this  point :  let  x  be  its  distance  from  S : 
the  moment  of  rotation  of  the  system  if  collected  in  that  point 
will  be 

(A+B+C+D-r&c.)*--; 
a 

which  must  by  hypothesis  be  equal  to 

Aa2+B63+Cc2+DdM-&c.- . 
a 

whence  we  obtain  for  the  value  of  x  in  all  cases 
Aa2+B62+CC2+&c.  . 

Aa+B6-fCc+&c.   '  ^      } 

and  for  the  integral  equation, 

*=7^-  <245> 

245.  We  have  seen  that  a  force  whose  direction  passes  through 
the  centre  of  gravity  of  a  body,  would  give  it  a  rectilineal  direc- 
tion only  ;  while  if  the  force  do  not  pass  through  the  centre  of 
gravity,  it  must  cause  it  to  revolve.  The  case' of  its  being  com- 
pelled to  revolve  on  a  fixed  axis  has  been  examined.  If,  however, 
the  body  have  no  fixed  point,  it  will  not  only  acquire  a  rotary, 
but  a  progressive  motion,  or  one  of  translation.  In  order  to  ex- 
amine the  manner  in  which  these  two  different  species  of  motions 
will  take  place  : — 

Let  ABCD  be  a  section  of  the  body,  passing  through  its  centre 
of  gravity,  G,  and  the  point  F  to  which  the  force  that  produces  the 
motion  is  applied ;  let  FH  represent  this  force  in  magnitude  and 
direction ;  from  the  centre  of  gravity  G,  draw  GI  perpendicular  to 
FH;  and  in  GI  produced  on  the  other  side  of  G,  take  GK  equal  to 


M     ^ — 


GI.     It  will  be  evident  that  the  condition  of  equilibrium  of  the 


236  ROTARY   MOTION  [Book  IV. 

system  will  not  be  changed  by  applying  two  forces,  KL  &  KM, 
to  the  point  K,  each  of  which  is  equal  to  the  half  of  FH,  provided 
they  be  parallel  to  FH,  and  act  in  opposite  directions.  The  force 
FH  may  then  be  considered  as  the  resultant  of  four  forces,  say  its 
two  halves  FN,  and  HN,  and  the  two  assumed  forces,  KL  and 
KM.  Of  these  four  forces,  two,  FN  and  KL,  concur  to  produce 
rectilineal  motion,  in  the  direction,  and  with  the  intensity  of  their 
resultant ;  the  direction  of  their  resultant  passes  through  the  point 
G,  is  parallel  to  their  direction,  and  equal  in  magnitude  to  the  ori- 
ginal force  FH.  The  other  two  forces,  HN  and  KM,  will  concur 
to  produce  a  rotary  motion,  around  an  axis  passing  through  the  same 
point  G  ;  and  this  axis  will  be  a  normal  to  the  plane  ABCD : 
hence  — 

If  a  force  be  applied  in  any  direction  to  a  body  which  is  free  to 
move,  it  will  cause  its  centre  of  gravity  to  describe  a  straight  line, 
parallel  to  the  direction  of  the  force  ;  and  will  communicate  to  the 
body  a  quantity  of  motion  equal  to  its  own  intensity.  The  velo- 
city may  of  course  be  found  by  dividing  the  force  by  the  mass 
of  the  body. 

If  the  force  do  not  pass  through  the  centre  of  gravity,  it  will, 
besides,  communicate  to  the  body  a  rotary  motion  around  an  axis 
passing  through  its  centre  of  gravity  ;  and  this  axis  will  be  a  normal 
to  the  plane,  passing  through  the  centre  of  gravity  and  the  di- 
rection of  the  force.  The  angular  velocity,  and  other  circum- 
stances of  the  rotary  motion,  may  be  computed  as  if  the  axis  of 
rotation  were  fixed.  For  it  will  be  at  once  seen,  that  if  we  were 
to  apply  to  the  body  a  force  in  a  direction  passing  through  the 
centre  of  gravity,  equal  in  magnitude,  and  contrary  in  direction,  to 
its  motion  of  translation,  we  should  destroy  this  motion  altogether, 
but  should  not  in  any  manner  affect  its  motion  of  rotation. 

When  a  body  revolves  upon  an  axis,  every  point  will  acquire 
a  centrifugal  force  proportioned  to  its  distance  from  the  axis. 
Hence  the  axis  of  rotation  will  have  a  constant  position,  only  when 
these  centrifugal  forces  are  in  equilibrio  around  it.  In  all  other 
cases,  the  axis  of  rotation  must  undergo  a  change.  A  homoge- 
neous sphere  may  revolve  permanently  upon  any  one  of  its  diame- 
ters. An  ellipsoid  of  revolution  may  revolve  permanently  around 
the  axis  of  the  generating  curve,  or  upon  any  one  of  its  equato- 
rial diameters;  but  upon  no  other  line,  for  the  several  points  that 
compose  this  solid  will  not  be  symmetrically  situated  in  respect  to 
any  other.  A  homogeneous  cylinder  may  revolve  permanently 
upon  its  geometric  axis,  or  upon  any  diameter  of  the  circle  that 
bisects  the  axis. 

246.  A  further  examination  of  the  properties  of  revolving  bo- 
dies, leads  to  a  remarkable  proposition  ;  the  investigation  of 
which  exceeds  the  limits  within  which  our  subject  is  restricted. 
It  is  as  follows  : 


Book  I  P.]  OJT  BODIES.  237 

In  any  body  whatever,  however  irregular,  there  are  three  axes 
of  permanent  rotation  at  right  angles  to  each  other,  upon  any  one 
of  which,  if  the  body  revolve,  the  centrifugal  forces  of  its  several 
points  will  be  in  equilibrio.  These  three  axes,  have  also  this  re- 
markable property,  that  the  moment  of  inertia  in  respect  to  them, 
is  either  a  maximum  or  a  minimum  ;  that  is  to  say,  is  greater  or 
less  than  if  the  body  revolved  around  any  other  line  as  an  axis. 

247.  When  a  body  has  a  double  motion,  of  rotation  and  trans- 
lation impressed  upon  it,  the  centre  of  gravity  will,  as  has  been 
seen,  move  forwards  with  uniform  velocity  ;  the  other  points  in 
the  body  will  move  with  velocities  that  are  continually  varying. 
Those  on  which  the  rotary  and  direct  motions  concur  for  an  in- 
stant, and  which  are  most  distant  from  the  axis  of  rotation,  will 
move  with  the  greatest  velocity ;  and  those  in  which  these  motions 
are  opposed,  will  move  with  the  least ;  and  some  of  the  points, 
most  distant  from  the  axis  of  rojtation,  will  actually  have  a  motion 
in  a  direction  contrary  to  that  of  the  centre  of  gravity. 

248.  There  will  also  be  a  point  in  the  system,  in  which  at  any 
instant,  the  progressive  and  rotary  motions  will  exactly  balance 
each  other.  This  point  is  called  the  Centreof  Spontaneous  Rotation. 
The  motion  of  the  body  far  any  instant  of  time,  may  be  considered 
asa  simple  rotary  motion  around  this  centre.    This  variety  in  the 
rate  at  which  the  different  points  of  a  body  move,  when  endued 
both  with  a  motion  of  rotation  and  translation,  has  no  effect  when 
the  body  moves  forward,  without  meeting  with  resisting  forces  ; 
but  when  these  act,  it  produces  marked  changes  in  the  direction 
and  circumstances  of  their  motion.     We  shall  have  occasion  to 
recur  to  these  circumstances  hereafter,  in  treating  of  motions  in 
resisting  media.     For  the  present  we  shall  confine  ourselves  to 
what  happens  when  a  body  is  rising  or  falling  in  the  air,  near  the 
surface  of  the  earth. 

The  resistance  it  meets  with  from  that  medium,  being  a  function 
of  the  velocity,  will  act  unequally  upon  opposite  sides  of  the  body, 
unless  the  rotation  be  around  a  vertical  line;  and  this  unequal  re- 
sistance will  produce  a  deviation  from  the  original  direction  of 
the  motion.  Hence,  a  heavy  body,  although  projected  vertically 
upwards,  rarely  or  never  falls  back  upon  the  exact  point  whence 
it  was  projected. 

There  are  various  instances  of  the  same  kind  to  be  met  with,  when 
the  air  or  other  resistances  act  upon  a  revolving  body,  and  the 
deviation  may  become  so  considerable,  as  to  bring  a  body  projected 
horizontally,  back  to  the  point  where  the  motion  began.  Thus,  if 
a  disk  of  metal,  such  as  a  coin,  placed  in  a  vertical  position,  upon 
a  plane  surface,  be  impelled  by  a  force  applied  to  either  extremity 
of  its  horizontal  diameter,  it  will  acquire  a  rotary  and  a  progres- 


238  ROTARY    MOTION  [Book  IV. 

sive  motion,  the  former  being  around  its  vertical  diameter;  the  un- 
equal action  of  the  air  upon  the  opposite  sides  of  the  vertical  axis, 
concurring  with  the  friction  of  the  plane  on  which  it  rests,  will 
cause  it  to  describe  a  series  of  re-entering  curves.  A  billiard  ball 
placed  on  a  plane  surface,  and  impelled  by  a  force  which  gives  its 
lower  side  a  motion  of  rotation  contrary  to  the  direction  of  its  cen- 
tre of  gravity,  will  have  its  progressive  motion  destroyed  by  the 
friction,  and  will,  afterwards,  by  virtue  of  the  rotary  motions  it  re- 
tains, roll  back  towards  the  place  whence  it  originally  set  out.  Cases 
of  a  similar  nature  are  too  frequent  and  familiar  to  need  enumera- 
tion. 

249.  The  general  expression,  (240),  for  the  value  of  the  mo- 
ment of  inertia,  may  be  applied  to  particular  cases  by  means  of 
the  calculus. 

Call  the  moment  of  inertia  S ;  let  dm  be  an  element  of  the 
figure  whose  moment  is  sought,  and  x  the  distance  from  the  axis 
of  rotation  ;  then 

8=fj*dm.  (246) 

To  adapt  this  to  individual  instances  : 

(1).  To  find  the  moment  of  inertia  of  a  straight  line,  revolving 
around  an  axis  perpendicular  to  itself,  and  passing  through  one  of 
its  extremities : 
then  dm  becomes  dx,  and 

S=, 
integrating 

84< 

and  when  x=a, 

S=£.  (247) 

(2).  To  find  the  moment  of  inertia  of  the  circumference  of  a 
circle,  in  respect  to  an  axis  passing  through  the  centre,  and  perpen- 
dicular to  the  plane  of  the  circle  : 

The  element  of  the  curve  being  ds,  and  its  distance  from  the 
axis  or  radius,  a, 

integrating 

and  when  s  becomes  the  whole  circumference,  or  2tfa, 

S=2*a3.  (248) 

(3).  To  find  the  moment  of  inertia  of  the  circumference  of  a 
circle  in  respect  to  a  diameter ;  let  x  and  y  be  the  ordinate  and 

abscissa;  the  element ds~ ;  its  distance  from  the  diameter  =  i/, 

and 


Book  IV.}  oy  BODIES.  239 

The  integral  of  ydx,  when  *  becomes  the  whole  circumference,  is 
equal  to  the  area  or  to  «ra2,  therefore, 

S=rf.  (249) 

(4).  To  find  the  moment  of  inertia  of  the  area  of  a  circle  whose 
radius  is  a,  in  respect  to  an  axis  passing  through  its  centre,  and 
perpendicular  to  its  plane  : 

Take  an  elementary  ring  whose  radii  are  z,  and  z-Mz,  the  area 
of  this  ring  will  be,  2«zdz,  and  its  moment  of  inertia,  considering 
it  as  the  circumference  of  a  circle,  will  be  2*a?dz,  by  case  (2)  ; 
hence, 

S 
Integrating 

a^fffl; 
and  when  z  —  a, 

S=i,r^a4.  (250) 

(5).  In  a  similar  manner  it  may  be  concluded,  that  the  moment 
of  inertia  of  the  area  of  a  circle  in  respect  to  one  of  its  diameters, 

S=i*a4.  (251) 

(6).  To  find  the  moment  of  inertia  of  a  solid,  formed  by  the 
revolution  of  a  curve,  in  respect  to  the  axis  of  rotation  : 

The  figure  being  symmetric  may  be  referred  to  no  more  than  two 
co-ordinates,  x  and  y  ;  take  for  the  element  the  solid  contained 
between  two  circles,  whose  distances  from  the  origin  of  the  co-or- 
dinates are  respectively  #,  and  x-\-dxt  the  moment  of  rotation  of 
the  element  will  be,  by  case  (3),,,ajjhj 


and  that  of  the  whole  solid, 

S=.i  tf/ydar.  (252) 

To  apply  this  formula  to  the  case  of  a  cylinder,  whose  length  is  6, 
and  the  radius  of  whose  base  is  a, 

S=itfa4&.  (253) 

In  a  cone,  the  radius  of  whose  base  is  o,  and  whose  altitude  is  6, 

S=TV<™'&.  (254) 

In  a  hemisphere, 

S—  T*j*a5.  (255) 

In  a  sphere, 

S=JLtfa5.  (256) 

250.  When  the  moment  of  inertia  of  a  body  in  respect  to  any 
axis  is  known,  it  is  easy  to  find  its  moment  of  inertia  in  respect 
to  any  other  axis,  provided  it  be  parallel  to  the  first. 


240 


ROTARY    MOTION 

First  let  the  given  axis  pass  through  the  centre  of  gravity  :  then 
let  GG'  be  this  axis,  and  CC'  another  parallel  to  it,  in  respect  to 


which  the  moment  of  inertia  is  sought  ;  let  an  element  of  the  sys- 
tem be  situated  at  M,  and  let  its  mass  be  dm  :  suppose  a  plane  to 
pass  through  M,  perpendicular  to  the  two  axes  GG',  and  CC',  and 
in  this  plane  draw  the  lines  MG',  MC,  and  join  CG'  ;  let  fall  MP 
from  the  point  M,  perpendicular  on  CG'.  Let  MG=r,  MC=y, 
then 


(sea) 

(3SS) 


in  the  right  angled  triangles  MCP,  and  MG'P,  ' 
MC3=MPa+CP2 

MPa=MG'2—  G'P3, 

and 

CPa=(G'P+CG')a, 

whence 

MC2=MG/3+2(CG'+G'P)+CG'2, 

and  S'  will  be  equal  to/dn?,  multiplied  by  the  second  member  of 
the  above  equation  ;  but  dm  multiplied  by  the  line  GP,  is  the  mo- 
ment of  inertia  of  the  element  dm,  in  respect  to  a  plane  passing 
through  the  centre  of  gravity  G;  and,  therefore,  fdm  multiplied 
by  the  varying  length  of  this  line  =0,  by  the  property  of  the  cen- 
tre of  parallel  forces,  §  25.  The  term  of  which  it  forms  a  part  is, 
therefore,  also  =0,  and  disappears  ;  hence,  if  we  call  the  distance 
between  the  centre  of  gravity,  and  the  point  C,  through  which  the 
second  axis  passes,  &; 

S'^fdm^+fdmk*  ;  (257) 

and  by  substituting  S,  for  its  value,  and  integrating  the  second 
term,  we  obtain 

(258) 


Book  IT.}  or  BODIES.  241 

The  moment  of  inertia  S",  in  respect  to  any  other  parallel  axis, 
whose  distance  from  the  centre  of  gravity  is  /b',  is 

S"=S-|-mfc/2;  (259) 

and  by  substituting  the  value  of  S,  obtained  from  the  last  equation, 
S"=S'-r-m(&'2— #»).  (260) 

251.  This  principle  may  be  applied  to  the  discovery  of  the 
moment  of  inertia,  in  cases  different  from  those  that  have  already 
been  investigated.  Thus : 

(6).  To  find  the  moment  of  inertia  of  a  circle,  in  respect  to  an 
axis  parallel  to  a  diameter  :  we  have  for  its  moment  of  inertia  in 
respect  to  that  diameter,  (251),  i<m*  : 

Call  the  distance  between  the  diameter  and  axis,  k ;  m  becomes 
the  area  of  the  circle,  or  tfa2,  and 

S=itfa4+*a3&.  (261) 

(7).  To  find  the  moment  of  inertia  of  a  solid  of  revolution,  in 
respect  to  an  axis  passing  through  its  vertex,  and  perpendicular  to 
the  axis  of  rotation  x : 

We  may  take  for  the  element  of  the  solid  in  this  case,  a  circle 
whose  radius  is  ?/,  and  whose  distance  from  the  axis  is  x.  Its 
moment  of  inertia,  in  respect  to  the  axis,  will  be,  §  247, 

and  to  the  vertex, 
and  its  mass  is 

whence 

$=i«fy*dx+«fx*tfdx.  (262) 

If  the  axis  pass  through  any  other  point,  whose  distance  from 
the  centre  of  gravity  is  r, 

S  =  A  «fy*dx+«r*fy2dx .  (263) 


31 


242  MOTION  OF  [Book   IF'. 

CHAPTER  III. 

Or  THB  MOTION  OF  PROJECTILES. 

252.  It  his  been  shown,  §  23S,  that  a  body  projected  upwards 
from  any  point  on  the  earth's  surface,  is  retarded  both  by  the  ac- 
tion of  the  force  of  gravity,  and  the  resistance  of  the  air.  If  pro- 
jected downwards  from  a  point  near  the  earth,  the  former  would 
accelerate  its  motion  with  its  whole  intensity  ;  the  latter  would 
still  act  to  retard  it.  But  if  it  be  projected  in  any  other  direction 
than  a  vertical  line,  the  resistance  of  the  air  continues  to  act  ac- 
cording to  the  same  law,  while  the  force  of  gravity  is  now  exert- 
ed in  a  direction  inclined  to  that  in  which  it  would  tend  to  move, 
under  the  action  of  the  projectile  force.  Let  us  in  the  first  place 
conceive  the  resistance  of  the  air  to  be  removed.  To  whatever 
point  of  the  body  the  projectile  force  is  applied,  its  centre  of 
gravity,  §  244,  will  acquire  a  motion,  in  a  direction  parallel  to 
that  of  the  force,  with  uniform  velocity.  The  force  of  gravity, 
within  the  limits  in  which  projectiles  move,  may  be  considered 
as  constant,  and  although  its  directions  are  all  converging,  no  im- 
portant error  can  arise  from  considering  it.  as  acting  always  par- 
allel to  itself.  The  case,  therefore,  becomes  the  same  as  that  we 
have  considered  in  Book  II.  Chap.  IV.  Hence  we  may  infer  : 

That  if  a  body  were  projected  from  a  point  near  the  surface  of  the 
earth  in  a  direction  that  is  not  a  normal  to  that  surface,  and  no 
other  accelerating  force  were  to  act  but  that  of  gravity,  it  would 
describe  a  parabola,  whose  diameters  are  normals  to  the  horizon- 
tal plane  : 

That  it  would  have  the  greatest  horizontal  range,  when  pro- 
jected at  an  angle  of  45  degrees  to  the  horizon  : 

That  it  would  rise  to  the  greatest  height,  the  more  nearly  its 
original  direction  approaches  to  the  vertical  ;  and  that  the  time 
of  flight  will  also  increase  with  the  increase  of  the  angle  of  eleva- 
tion. 

If  we  call  h  the  height  whence  the  body  would  have  fallen  to 
acquire  the  initial  velocity  v  ;  d  the  horizontal  distance  to  which 
the  projectile  is  carried  ;  i  the  angle  of  inclination  ;  and  y  the  force 
of  gravity,  we  obtain  from  (72)  and  (61), 


,  (264) 

which  is  a  maximum  when 

*  t=45°; 

in  which  case  the  equation  becomes 


Book  Ifr.}  PKOJKCTILES.  243 

&=*.=  *-  (265) 

g       32 
If  we  suppose  the  initial  velocity  to  be  2000  feet  per  second, 

d=  125000  feet, 
or  nearly  twenty  four  miles. 

253.  When  projectiles  move  with  but  small  velocity,  the  dis- 
crepancy between  the  parabolic  theory,  and  what  is  found  to  oc- 
cur in  practice,  is  but  small;  but  with  increasing  velocities,  as 
the  air's  resistance  increases  in  a  higher  ratio  than  the  velocity, 
this  discrepancy  becomes  very  great.  The  general  effect  of  a  re- 
sisting medium  may  be  thus  investigated.* 

Let  g-R  represent  the  resistance  of  the  medium,  which  will  be 
exercised  in  a  direction  contrary  to  that  of  ds,  the  element  of  the 
curve  at  any  given  point.  Refer  the  curve  to  two  axes,  one  ver- 
tical, t/,  the  other  horizontal,  #,  both  passing  through  the  point  of 
projection. 

If  we  decompose  the  resistance,  g-R,  into  two  forces  parallel  to 
the  two  axes,  that  parallel  to  x  will  be 

**£' 

that  parallel  to  t/, 


and  these  being  retarding  forces,  will  have  the  negative  sign.  The 
first  will  express  the  whole  disturbing  force  in  the  direction  paral- 
lel to  x  ;  but  in  the  direction  parallel  to  ?/,  the  force  of  gravity  acts 
also  :  hence  the  whole  forces  parallel  to  the  two  co-ordinates  be- 
come 


and 

°     ^       ds ' 

From  the  general  equations  of  curvilinear  motion,  §  62,  we 
have 

and 

whence  we  have 

dx       _d*x 

*Venturoli,  Vol.  1.  p.  95. 


244  MOTION  OP  [Book  IV. 

dy_d*y 

—g—g*  Sf—jr- 

The  velocity  in  the  direction  of  a:  is  constant  by  (236),  there- 
fore, by  taking  the  differential  of  both,  and  substituting  in  the  se- 
cond the  value  of  d2/,  we  obtain 


and  taking  the  differential  of  the  second,  we  have 


eliminating  dt  and  dzt  from  this  equation,  we  have  for  the  equation 
of  the  curve, 


Whence,  if  the  law  of  the  resistance  were  known,  we  might  pro- 
ceed to  determine  the  nature  of  the  curve. 

Assuming  that  the  resistance  of  the  air  varies  with  the  square 
of  the  velocities,  it  has  been  attempted  to  ascertain  the  curve 
which  a  projectile  would  describe.  Its  differential  equation 
has  not,  however,  yielded  to  the  integral  calculus  ;  and  were  the 
resistance  to  bear,  as.  we  hereafter  shall  see  that  it  does,  a  more 
complex  relation  to  the  velocity,  even  the  differential  equation 
would  become  difficult  to  obtain.  Although  this  curve  does  not 
satisfy  the  actual  conditions  of  the  resistance,  it  will  still  be  a 
matter  of  interest  to  state  what  is  known  in  respect  to  it,  from  its 
differential  equation,  and  from  a  method  of  determining  points 
through  which  it  passes,  that  has  been  thence  deduced.  While 
the  two  branches  of  the  parabola,  measured  from  the  horizontal 
plane,  are  similar  and  equal  to  each  other,  and  have  equal  bases  ; 
the  two  branches  of  the  curve  that  would  be  described  under  the 
action  of  a  projectile  force,  the  attraction  of  gravitation,  and  a 
resistance  varying  with  the  squares  of  the  velocity,  are  unequal  ; 
the  base  of  its  ascending  branch  being  longer  than  that  of  the  de- 
scending branch. 

In  the  parabola,  the  times  of  describing  the  two  branches  are 
equal  ;  and  the  velocity  of  the  projectile,  when  it  again  reaches 
the  horizontal  plane  passing  through  the  point  of  projection,  is 
equal  to  the  initial  velocity.  In  the  curve  under  consideration, 
the  time  of  describing  the  ascending  branch  is  less  than  that  of 
describing  the  descending  branch,  and  the  final  velocity  is  less 
than  the  initial  velocity,  and  may  become  constant. 

254.  The  most  important  application  of  our  subject  is  to  what 
are  styled  military  projectiles.  These  are  impelled,  by  the  ex- 
plosion of  gunpowder  from  instruments  having  cylindrical  cavi- 
ties, and  which  are,  in  general  terms,  called  Pieces  of  Ordnance. 

When  gunpowder  is  inflamed,  the  whole  is  converted  into  an 


Book  IF'.]  PROJECTILES.  245 

elastic  fluid,  whose  bulk  is  many  times  greater  than  that  of  the 
solid  whence  it  is  derived,  and  whose  elastic  force  is  enhanced  by 
the  high  temperature  at  which  it  is  generated. 

By  the  experiments  of  Hutton,  to  which  we  shall  presently 
refer,  gunpowder  would  appear  to  expand  itself  with  a  velocity 
of  5000  feet  per  second,  and  to  exert  a  force  at  least  2000  times 
as  great  as  the  pressure  of  the  atmosphere.  This  deduction  was 
obtained  from  its  action  upon  balls,  fired  from  pieces  of  ordnance. 
The  weight  of  these  is  small  compared  with  the  force  exerted  by 
the  gunpowder,  and  their  motion  is  at  first  but  little  resisted ; 
hence  they  are  set  in  motion  before  the  whole  of  the  gunpowder 
is  fired,  and  do  not  sustain  its  entire  action.  It  is  even  well 
established  by  experience,  that  no  inconsiderable  quantity  of  the 
gunpowder  isjoften  blown,  uninflamed,  out  of  the  gun,  by  the  ex- 
plosion of  the  rest. 

255.  Count  Rumford  made  experiments  in  another  manner. 
The  gunpowder  was  in  very  small  quantities,  and  was  placed  in 
a  small  vessel  of  wrought  iron,  without  a  vent.  The  ignition 
was  accomplished,  by  heating  this  vessel  red  hot  from  without. 
In  this  way  the  escape  of  elastic  fluid,  that  takes  place  in  pieces 
of  ordnance  from  the  vent,  was  avoided.  T^he  resistance  to  be 
overcome  was  a  weight  of  great  amount,  when  compared  with 
the  quantity  of  gunpowder,  being  a  brass  battering  cannon  of  the 
size  called  a  twenty-four  pounder  ;  the  cascable,  or  spherical  knob 
that  terminated  the  breech  of  this  gun,  was  fitted  to  the  vessel 
containing  the  gunpowder,  by  grinding,  in  such  a  manner  as  to 
make  an  air-tight  joint.  Experimenting  with  this  apparatus, 
Rumford  inferred  that  the  force  exerted  by  confined  gunpowder, 
was  equivalent  to  the  pressure  of  100000  atmospheres. 

This  would,  at  first  sight,  appear  to  exceed  the  limits  of  credi- 
bility ;  but  there  are  cases  in  which  gunpowder  does  actually  ex- 
ert a  force  far  greater  than  it  could,  were  its  energy  no  more 
than  was  inferred  by  Hutton.  As  instances  of  this  kind,  may  be 
cited  what  happens  in  the  blasting  of  rocks,  and  in  military 
mines,  particularly  in  the  effect  produced  by  the  latter,  that  is 
called  the  globe  of  compression.  In  this,  besides  throwing  up  a 
large  conoid  of  earth,  the  gunpowder  shakes  the  ground  to  a 
considerable  distance  around  it,  acting  with  sufficient  energy  to 
overturn  and  destroy  walls  cf  solid  masonry.  There  being  this 
great  difference  in  the  action  of  gunpowder,  when  it  is  exerted 
against  a  body  that  is  easPy  set  in  motion,  and  when  it  is  closely 
confined  ;  it  will  be  at  ence  seen  that  great  dangers  may  arise, 
when,  from  accident  or  intention,  the  projectile  to  be  launched 
from  a  piece  of  ordnance  is  resisted  in  its  motion.  Thus,  if  the 
muzzle  of  a  gun  be  inserted  in  water,  if  a  portion  of  air  be  left 


246  MOTION  OF  [Book  IV. 

between  a  wad  and  the  rest  of  the  charge  ;  if  the  projectile  be 
of  a  hard  material,  and  of  such  a  shape  that  it  may  strike  before 
it  issues  from  the  piece  :  in  all  these  cases,  the  strength  of  the 
material  of  which  the  piece  is  formed  may  not  be  sufficient  to 
resist  the  accumulation  of  force,  and  bursting  may  be  the  conse- 
quence. So,  also,  if  the  wad  be  of  a  cohesive  material,  such  as 
tarred  yarn  ;  and  particularly  when  it  is  so  large  as  to  enter  the 
piece  with  difficulty,  similar  consequences  may  ensue.  To  the 
latter  cause  we  may  with  certainty  attribute  the  bursting  of  guns 
in  the  navy  of  the  United  States,  and  to  the  frequent  loss  of  them 
in  the  proof.  We  have  ourselves  witnessed  a  case  in  the  proof 
of  guns,  where  the  balls  made  their  way  through  the  sides  of  the 
piece,  and  large  portions  of  the  wad  remained  sticking  to  the 
bore  in  front  of  them. 

256.  The  theory  of  projectiles,  as  has  been  seen,  cannot  be 
completed  by  the  aid  of  mathematics,  and  it  hence  becomes  ne- 
cessary, in  order  that  the  practice  of  gunnery  may  become  sure, 
that  experiment  be  made  upon  a  considerable  variety  of  pieces  of 
ordnance,  with  projectiles  of  various  descriptions,  at  different  an- 
gles of  elevation,  and  with  varying  charges  of  gunpowder.     The 
best  general  experiments  of  this  nature  have  been  made  by  Hut- 
ton  and  Robins.  In  addition,  the  artillery  services  of  several  Eu- 
ropean nations,  and  particularly  that  of  France,  are  in  possession  of 
manuscript  tables  of  the  effects  of  their  several  descriptions  of 
ordnance,  derived  from  the  experiments  that  are  annually  making 
at  their  schools  of  practice.     The  experiments  of  Hutton,  Ro- 
bins, and  some  of  less  extent  made  by  Count  Rumford,  being 
alone  accessible,  we  shall  be  compelled  to  confine  ourselves  to 
them  :  they  are,  however,  sufficient  for  the  foundation  of  a  cor- 
rect theory,  on  which  a  sound  practice  may  be  established. 

257.  The  first  point  to  be  ascertained  in  experiments  on  mili- 
tary projectiles,  is  the  initial  velocity,  which  serves  as  the  basis 
of  all  the  other  investigations.      In  the  older  experiments,  this 
was  done  by  pointing  the  piece  vertically  upwards,  and  observing 
the  time  that  elapsed  between  the  discharge  and  the  return  of 
the  ball  to  the  ground  ;  half  of  this  was  assumed  as  the  time  of 
descent,  and  the  velocity  calculated  by  the  formula,  (231) 

v=32t. 

This  method  was  found  to  give  a  velocity,  that,  if  small, 
agreed  tolerably  well,  in  the  ranges  calculated  from  it,  with  the 
parabolic  theory.  But  at  the  usual  velocities  of  military  pro- 
jectiles, it  was  found  to  give  results  that  varied  very  much  from 
that  theory. 


Book  IV.  PROJECTILES.  247 


258.  Robins  next  invented  an  apparatus  styled  by  him  the  Bal- 
listic Pendulum.  This  is  composed  of  a  large  mass  of  wood,  sus- 
pended by  a  rod,  from  centres  on  which  it  is  free  to  vibrate. 
The  ball  is  fired  against  this,  and  the  velocity  communicated  to 
it  being  measured,  that  of  the  ball  may  be  easily  determined. 
The  whole  of  the  ball's  motion,  provided  it  did  not  pass  through 
the  wooden  mass,  would  be  communicated  to  the  latter  ;  or  rather 
the  ball  and  it  would  go  on  with  a  common  velocity,  whence  the 
quantity  of  motion  could  be  determined,  by  multiplying  the  ve- 
locity by  the  joint  mass  of  the  ball  and  the  pendulum.  This  pro- 
duct, divided  by  the  mass  of  the  ball,  would  give  the  velocity  of 
the  latter. 

So  far,  the  principle  is  simple  :  a  difficulty,  however,  arises  in 
the  determination  of  the  velocity  of  the  pendulum  ;  this  does 
not  go  on  with  uniform  motion,  but  rises  in  a  circular  arc,  until 
the  force  of  gravity  checks  its  motion,  and  causes  it  again  to  re- 
turn to  the  point  whence  it  set  out.  But  if  the  arc  be  known,  the 
doctrine  of  the  motion  of  pendulums,  which  will  be  explained 
hereafter,  enables  us  to  calculate  the  velocity  that  is  destroyed  in 
describing  it. 

Another  difficulty  arises  from  the  fact  that  the  pendulum  will 
describe  different  arcs,  according  to  the  greater  or  less  distance  of 
the  point  the  ball  strikes,  from  the  centre  of  motion  ;  and  it  will 
also  be  obvious  from  §  243,  that  if  the  ball  be  not  fired  against 
the  centre  of  percussion,  a  part  of  the  force  will  be  expended 
upon  the  axis  on  which  the  pendulum  hangs.  This  difficulty  was 
removed  by  Hutton,  who  investigated  a  theorem  whence  the  ve- 
locity could  be  calculated,  when  the  position  of  the  centres  of  gra- 
vity and  of  percussion,  and  that  of  the  point  where  the  ball  strikes, 
are  known  j  his  formula  is  as  follows  :  viz. 

v=5.6727  g  c 

in  which  6  is  the  weight  of  the  ball, 

p     the  weight  of  the  pendulum, 

g     the  distance  between  the  centre  of  gravity  and  the 
point  of  suspension, 

0  the  distance  between  the  centre  of  percussion  and 
the  point  of  suspension, 

1  the  distance  from  the  point  struck  by  the  ball  to  the 
point  of  suspension, 

c      the  chord  of  the  arc  described  by  the  pendulum,  and 
r      its  radius. 

The  position  of  the  centre  of  gravity  was  determined  experi- 
mentally, by  balancing  the  pendulum  upon  an  edge,  according  to 
the  principles  of  §  114. 
The  quantity  o,  was  ascertained  by  suspending  the  pendulum 


248  MOTION  or  [Book  IV. 

and  finding  the  time  of  its  vibration  in  very  small  arcs,  whence 
the  length  can  be  calculated,  as  will  be  hereafter  explained  in 
treating  of  pendulums. 

The  chord  c,  was  measured,  by  attaching  to  the  pendulum  a 
graduated  tape,  which  was  drawn  out  by  the  motion  of  the  pen- 
dulum, from  between  two  steel  edges. 

Hutton  also  determined  the  initial  velocity  of  the  ball  from  the 
recoil  of  the  gun,  by  means  of  the  principle  of  inertia  §  39.  By 
this  it  is  apparent  that  the  gunpowder  must  communicate  to  the 
gun  as  much  motion  as  it  is  capable  of  giving  to  the  ball,  but  in  a 
contrary  direction.  The  gun  rendered  heavier  by  additional 
weights,  was  suspended  in  the  same  manner  as  the  pendulum,  and 
its  velocity  of  recoil  ascertained  in  the  manner  that  has  been  de- 
scribed in  the  last  section. 

Experimenting  in  these  ways,  it  was  found  that  the  greatest 
initial  velocity  of  a  military  projectile,  does  not  much  exceed  two 
thousand  feet  per  second. 

259.  As  the  elastic  fluid  generated  by  the  firing  of  gunpowder, 
acts  upon  the  ball  during  the  whole  time  of  its  continuance  in 
the  piece,  tending  to  expand  with  a  velocity  of  5000  feet  per 
second,  while  the  ball  does  not  acquire  a  velocity  much  greater 
than  2900  feet :  the  latter  must  be  accelerated  during  the  whole 
of  its  continuance  in  the  piece,  provided  its  velocity  does  pot  be- 
come so  great,  that  the  resistance  of  the  air,  and  the  friction  to 
which  it  is  subjected,  are  sufficient  to  render  its  velocity  constant. 
This  acceleration  is  not,  however,  uniform,  but  becomes  less  and 
less  as  the  ball  moves  forward,  for  the  following  reasons: 

(1.)  The  elasticity  of  the  fluid  proceeding  from  the  inflamed 
gunpowder,  decreases  with  the  increase  of  the  space  it  occupies. 

(2.)  The  elasticity  depends  in  part  upon  the  temperature,  and 
this  will  decrease  as  the  gas  expands,  and  also  by  the  conducting 
power  of  the  piece. 

($.)  Forces  of  this  nature  act  with  more  intensity  upon  bodies 
at  rest  than  upon  bodies  in  motion. 

(4.)  The  ball  is  resisted  by  the  air,  a  retarding  force  that  in- 
creases in  a  higher  ratio  than  does  the  velocity  of  the  ball,  and 
which  may  finally  render  the  latter  constant.  See  §  239. 

Thus,  although  an  increase  in  the  length  of  the  bore  does,  in 
all  cases  that  can  occur  in  practice,  increase  the  initial  velocity, 
still  it  does  not  do  so  in  the  same  ratio  in  which  the  length  of  the 
bore  is  increased. 

The  experiments  of  Hutton  showed  that  this  increase  was  in 
a  ratio  not  as  great  as  that  of  the  square  root  of  the  length  of  the 
piece,  but  in  a  ratio  greater  than  that  of  the  cube  root ;  and 
that  if/  be  the  length  of  the  piece,  the  ratio  is  almost  exactly 

/£.  (266) 


PROJECTILES.  249 

260.  The  velocities  communicated  to  balls  of  equal  weights, 
in  pieces  of  the  same  length,  by  unequal  quantities  of  powder, 
were  as  the  square  roots  of  the  quantities  of  powder. 

The  velocities  communicated  to  balls  of  different  weights,  in 
pieces  of  equal  lengths,  by  unequal  quantities  of  pov/der,  were 
inversely  as  the  square  roots  of  the  weights  of  the  balls. 

If  p  be  the  quantity  of  gunpowder,  6  the  weight  of  the  ball, 
and  »»  a  co-efficient,  constant  for  all  pieces  of  similar  form, 


..  (267) 

This  co-efficient,  m,  may  be  safely  taken  at  1600  feet  in  most  of 
the  cases  that  occur  in  practice. 

A  remarkable  consequence  follows  from  the  above  formula  : 
If  the  weight  of  the  ball  be  increased,  by  using  solid  instead  of 
hollow  balls,  or  making  them  of  denser  substances,  the  velocity, 
all  other  things  being  equal,  does  not  decrease  more  rapidly  than 
in  the  inverse  ratio  of  the  square  root  of  the  weight;  and  hence 
a  heavy  ball  will  have  a  greater  quantity  of  motion  than  a  lighter 
one,  projected  from  the  same  piece,  with  equal  quantities  of 
powder. 

261.  The  resistance  of  the  air  was  also  carefully  examined  by 
Hutton  ;  we  have  not  space  to  enter  into  the  detail  of  his  expe- 
riments :  it  is  sufficient  to  state,  that  at  all  velocities,  from  300 
to  1100  feet  per  second,  he  found  that  the  resistances  might  be 
thus  expressed  : 

r=mv2+nv.  (268) 

In  this  formula,  v  is  the  velocity,  and  m  and  n  constant  co-effi- 
cients, which  for  balls  of  the  diameter  of  2  inches,  are, 
m=     .00002665, 
n=—  -.00388. 

The  resistance  to  balls,  is  found  to  vary  with  the  squares  of 
their  diameters.  The  formula  for  any  other  diameter  (a)  will  there- 
fore be 

r=(mv3__W7))  a*t  (269) 

in  which 

m  =  .  000007657, 
w=.000175. 

Calculations  founded  upon  this  formula,  are  found  difficult  in 
practice  ;  and  hence  a  co-efficient  is  chosen  from  the  experiments, 
which  will  affect  only  the  square  of  the  velocity,  at  such  velocities 
as  are  most  frequently  employed  in  practice.     Call  this  co-effi- 
cient c,  the  whole  resistance  to  the  body  expressed  in  Ibs.  will  be 
ct>3  a3, 
32 


MOTION  OF  [Book 

which  will  be  equal  to  the  value  of  the  part  of  the  retarding  force 
in  §  239,  represented  by  Ar,  multiplied  by  the  weight  of  the 
body,  or 


whence 

«*=^,  (270) 

and  gk  is  the  measure  of  the  retarding  force. 

Let  t£  be  the  initial  velocity,  u  the  velocity  remaining  after 
moving  through  a  distance,  x  ;  in  order  to  find  the  value  of  x  : 

By  the  general  formula  of  variable  motion,  (53) 
.     do     »  dv 

f=sV=di=-fo   ; 

and  substituting  x  for  *, 
t  dv 

f=dx~i 
whence 

v  dv=fdx  ; 

substituting  the  value  of/  or  gk2,  and  considering  that  the  force 
is  a  retarding  one,  we  obtain 

A  J*  <"'"' 

—  v  dv=g  <&•"*£•  ; 

and 

to         —  dv 

*"*Z3  •  — 

Integrating,  and  considering  that  when  ar=o,  0=u,  we  obtain 


To  give  this  a  more  convenient  form,  and  to  adapt  it  to  the  case 
of  balls  of  cast  iron  : 

The  weight  of  a  cubic  inch  of  cast  iron  is  4.3  oz.,  hence 

tc=.5236a3  X  4.3=2.25a3 

in  ounces  avoirdupois,  g1  the  measure  of  the  force  of  gravity,  is 
32  feet,  and  the  co-efficient,  c,  suited  to  velocities  of  from  1200  to 
1400  feet  per  second  is  .000007657 ;  hence  we  have 

w 

—  =  58Ha; 
gca* 

multiplying  this  co-efficient  by  2.30258,  in  order  that  we  may 
substitute  common  for  hyperbolic  logarithms,  we  obtain 

*= 13380.  log.  £;  (272) 


Book  /F".]  PROJECTILES.  251 


from  this  we  obtain  for  the  values  of  u  and  0, 


A  still  more  convenient  form  may  be  given  to  this,  by  consi- 
dering that  if  D  be  the  density  of  the  ball, 

w=.5236a3D; 

which,  taking  atmospheric  air  as  the  unit,  gives  us 
20  Da  u 

—9—         log.-;  (274) 

whence 


log.  v— log.  u  —     Q  |. 

The  calculation  of  the  two  last  formulae  could  be  made  more 
easy,  if  they  had  the  form 

log.  t,=log.  r-flog.  m      ) 

log.  v=log.  u — ^log.  m.      )  '  v       / 

If  we  make 

20  Da 


this  condition  may  be  satisfied  by  finding  a  constant  number,  by 

which  if  -  be  multiplied,  it  will  give  a  logarithm,  which,  within 
c 

the  usual  limits  of  the  range  of  military  projectiles,   roes   not 
differ  much  from 


the  fraction  0.43429448,  is  one  that  is  best  suited  for  this  pur- 
pose ;  hence,  in  the  preceding  formulse, 

log.    n»=-X  0.43429448.  .  (277) 

To  find  the  time  employed  in  describing  the  space  *,  we  have 

(53) 

dx 


from  which  may  be  obtained,  by  substitution  and  integration,  em- 
ploying afterwards  the  quantities  c  and  MI,  according  to  the  prin- 
ciples just  laid  down, 

«=;     (m-l);  (278) 


253  MOTION  OP  [Book  IV. 

if  the  initial  velocity,  and  the  time  of  flight  be  given,  we  have  for 
the  value  of  the  space  x  from  (277) 


and  for  m,  from  the  previous  equation, 

,n=—  +1.  (279) 

In  order  to  determine  the  angle  of  projection  at  which  a  projec- 
tile with  a  given  velocity  will  strike  the  horizontal  plane  ai  a  given 
distance  ;  we  must  consider,  that  the  projectile,  as  soon  as  it 
leaves  the  piece,  is  acted  upon  by  the  force  of  gravity,  by  which, 
if  we  abstract  from  the  air's  resistance,  it  will  in  a  time,  /,fall  through 
a  space,  s1  whose  value  is  (231) 
s=16  t3. 

If  we  consider  the  space,  ar,  which  the  ball  passes  through,  as 
coinciding  with  the  horizontal  distance,  (and  this  would  be  the 
case  if  the  ball  described  a  straight  line)  we  have,  for  the  tangent 
of  the  angle  of  elevation  f  , 

16/3  . 
Tan.  i=—  —  (280) 

and  substituting  the  value  of  t  from  (278) 

16 
Tan.t  =  ^  c2    (m—  I)2;  (281) 

from  which  may  be  obtained,  by  substituting  the  value  of  w, 

16      /  a?       \ 
Tan.  i=-y     (-+*)  5  (282) 

whence  we  have 


tan. 


262.  To  enable  us  to  apply  these  formulae  to  practice:  Cast 
iron  has  a  density,  in  terms  of  water,  as  the  unit  of  7.4,  as  deter- 
mined by  Hutton,  who  also  assumes  that  the  relation  of  the  den- 
sities of  water  and  air  are  as  1000:  If  ;  hence  we  have  for  the 
density  of  cast  iron  in  terms  of  air,  6054. 

We  have  next  to  ascertain  the  diameters  of  the  more  usual  pro- 
jectiles, in  fractions  of  feet.  These,  as  will  he  hereafter  more 
particularly  described,  are,  in  the  case  of  cannon,  distinguished 
by  their  weights.  Their  denominations  and  dimensions,  are  as 
follows  : 


Book 


PROJECTILES. 


253 


.Ball  of 
42  Ibs. 
32  Ibs. 
24  Ibs. 
18  Ibs. 
12  Ibs. 

9  Ibs. 

6  Ibs. 

4  Ibs. 


Diameter  in  inches. 
7.018 
6.410 
5.823 
5.292 
4.623 
4.200 
3.668 
3.204 


Diameter  in  feet. 
0.5848 
0.5342 
0.4852 
0.4410 
0.3852 
0.3500 
0.3057 
0.2670 


fed 

$ 

0 

*ji 

H 

LOGARITHMS  OF 

LOGARITHMS  OF 

CO 

s  « 

H 

H-l 

to 

20  Da 

1 

»J 

<t 

5 

-X  0.43439449 

PO 

Q  ~ 

Q 

9 

c 

42  Ibs. 

0.5848^ 

3.8958373 

5.7420470 

32  Ibs. 

0.5342 

3.8565338 

5.7813505 

24  Ibs. 

0.4852 

3.8147507 

5.8231336 

18  Ibs. 

0.4410 

3.7732685 

5.8646158 

12  Ibs. 

0.3852 

"    6054' 

3.7145162 

5.9233681 

9  Ibs. 

0.3500 

3.6728979 

,  5.9649864 

6  Ibs. 

0.3057 

3.6151253 

6.0127590 

4  Ibs. 

0.2670  .. 

3.5553412 

6.0825431 

The  pieces  of  ordnance  most  frequently  employed  in  the  prac- 
tice of  artillery  are,  Cannon,  Mortars,  Howitzers,  and  Carron- 
ades.  The  diameters  of  the  bore  of  all  these  different  pieces 
are  called  their  Calibres. 

Cannon  vary  from  twelve  to  twenty-four,  or  even  more  calibres 
in  length  ;  they  are  moveable  upon  two  axles  that  are  prolongations 
of  the  same  cylinder,  whose  axis  is  a  normal  to  the  vertical  plane 
passing  through  the  axis  of  the  piece,  and  which  are  called  Trun- 
nions ;  by  these  they  are  attached  to  a  carriage,  on  which  they  may 
be  moved  from  place  to  place.  The  form  of  the  carriage  admits  them 
to  be  elevated  or  depressed,  a  few  degrees  above  or  below  the 
horizontal  plane.  They  are  distinguished,  according  to  their  use, 
into  Navy,  Battering,  Garrison,  and  Field  Guns  or  Pieces. 

The  axis  of  the  trunnions  is  usually  in  the  vertical  plane  pas- 
sing through  their  centre  of  gravity,  but  intersects  it  below  that 
point. 

Mortars  are  short  pieces  of  ordnance  whose  trunnions  are  at 
their  breech.  By  these  they  are  adapted  to  beds.  In  the 
English  service,  they  are  permanently  fixed  to  these  beds,  at  an 
angle  of  45°  with  the  horizon.  In  the  French,  and  in  our  land 
service,  they  are  moveable  upon  their  trunnions,  so  that  they  may 
be  fired  at  any  angle  of  elevation. 

Howitzers  have  a  form  similar  to  that  of  mortars,  but  have 
their  trunnions  situated  like  those  of  cannon.  They  are  mounted 


254  MOTION  OF  [Book 

on  the  same  kind  of  carriages  as  cannon,  but  as  they  are  shorter, 
are  capable  of  greater  angles  of  elevation. 

Carronades  are  short  cannon,  used  almost  wholly  in  the  naval 
service.  They  have  no  trunnions,  but  are  connected  with  their 
carriages  or  slides  by  iron  straps,  passed  through  a  cylindrical 
opening,  made  in  a  piece  cast  on  their  lower  sides  for  the  purpose. 
All  these  pieces  of  ordnance  are  now  cast  solid,  and  the  cylindri- 
cal cavity  that  contains  the  charge  is  cut  out  by  a  rotary  motion  ; 
whence  it  is  called  the  Bore. 

The  bore  of  mortars,  howitzers,  and  carronades,  is  made  of 
smaller  diameter  towards  the  breech  ;  thus  assuming  the  shape  of 
two  cylinders  united  by  a  portion  of  a  spherical  surface.  The 
smaller  part  of  the  bore  is  of  such  length  as  to  receive  the  maxi- 
mum service  charge  of  gunpowder,  and  is  called  the  Chamber. 
Some  cannon  also  have  chambers,  as  have  the  better  description  of 
small  arms.  The  formulae  that  have  been  given  above,  are  appli- 
cable to  cannon,  howitzers  and  carronades,  but  not  to  mortars, 
unless  fired  at  small  angles  of  elevation  ;  a  case  that  rarely  occurs 
in  practice ;  for  it  is  only  at  small  angles  of  elevation  that  the 
actual  path  of  the  projectiles  approaches  nearly  to  the  distance, 
measured  in  a  straight  line. 

The  distance  called  Point  Blank,  in  the  English  service,  is 
estimated  from  the  position  of  the  piece  to  the  point  at  which 
the  curved  path  of  the  projectile  intersects  the  horizontal  plane 
on  which  the  carriage  is  placed  ;  the  axis  of  the  piece  being  also 
horizontal.  The  term,  de  But  en  Blanc,  of  the  French,  fre- 
quently translated  point  blank,  supposes  the  line  of  sight  to  be 
directed  in  a  horizontal  line  over  the  highest  points  of  the  breech 
and  muzzle.  As  these  lie  on  the  surface  of  a  cone,  the  line  of  sight 
is  inclined  to  the  axis  of  the  piece,  the  Litter  of  which  is,  in  con- 
s.equcnce,  inclined  upwards  from  the  horizon  ;  the  path  of  the 
projectile  being  a  curve,  will  cut  the  line  of  sight  in  rising,  at  no 
great  distance  from  the  mouth  of  the  piece,  and  again  descending, 
will  cut  it  a  second  time,  at  a  distance  that  will  vary  with  the  ve- 
locity, arvd  the  angle  of  inclination.  The  distance  from  the  piece, 
to  the  point  where  the  path  of  the  projectile  cuts  the  line  of  sight 
the  second  time,  is  the  distance  de  But  en  Blanc.  To  strike  a 
point  at  this  distance,  the  piece  is  aimed  directly  at  the  object,  by 
means  of  sights,  cut  into  the  breech,  and  the  swell  of  the  muzzle; 
and  the  method  will  be  as  accurate  as  any  method  in  gunnery  can 
well  be,  whether  the  point  aimed  at  lies  in  the  same  horizontal 
plane  with  the  piece,  or  be  elevated  or  depressed  in  respect  to 
that  plane,  within  the  limits  of  elevation  and  depression,  that  the 
carriage  of  the  piece  will  admit. 

In  order  to  extend  the  advantages  of  the  method  of  direct  aim, 


Book  IV.]  PROJECTILES.  255 

to  objects  more  remote  than  the  distance  de  But  en  Blanc,  the 
French  artillerists  adapt  a  moveable  sight,  called  the  Hausse,  to 
the  breech  of  the  piece  ;  when  this  is  raised,  and.  the  line  of 
sight  again  directed  to  the  object,  the  axis  of  the  piece  will  be 
more  elevated  than  before,  and  the  horizontal  range  in  consequence 
greater.  When  the  abject  is  nearer  than  the  distance  de  But 
en  Blanc,  the  original  or  natural  line  of  sight  must  be  directed  to 
a  point  below  the  object. 

The  calculation  of  the  angles  of  elevation  corresponding  to  the 
natural  line  of  sight,  and  to  that  given  by  known  elevations  of  the 
Hausse,  may  be  ascertained  as  follows  : 

The  angle  of  elevation  will  be  equal  to  the  least  angle  of  a 
right-angled  plane  triangle,  of  which  one  side  is  the  difference  be- 
tween the  radii  at  the  breech,  and  at  the  muzzle,  and  the  other 
the  length  of  the  piece  :  hence,  if  R  and  r,  be  the  respective 
radii,  and  /  the  length, 

.     R—  r 

tan.  i=  :  ' 

if  now  the  Hausse  be  used,  let  M  be  the  height  to  which  it  is  ele- 
vated, 

.     R+H—  r 
tan.  i=  -  -  --  . 

The  application  of  these  principles  and  formulae  to  practice, 
may  be  illustrated  by  the  following 

EXAMPLES. 

(1).  A  24  pound  ball  is  projected  with  a  velocity  of  \2QQfeet 
per  second,  —  required  the  velocity  it  will  have,  after  passing 
through  a  distance  of  900  feet  ? 

The  formulae  are, 

log.  v=log.  u  —  log.  m,  (276) 

log.  m—  —  X  0.4343  ;  (277) 

and 

w=1200  feet  ; 

x—  900  feet  ; 
log.     (w=1200,)    .         .  3.0791812 

log.  ix  0.4343  5.8231336 

3    c 
log.      (#=900,)     .          .  2.9542425 


log.  (*X  0.4343-  log.  w)  8.7773761  No.    0.0598930 

log.  (i>=1045)  -..  ;'        •          ,          .         .         3.0192882 


256  MOTION  or  [Book  IV. 

(2).  A  24-pound  ballis  fired  at  an  object  distant  9QQ  feet,  and 
will  require  there  a  velocity  of  1000  feet  to  produce  the  proper 
effect,  —  required  the  initial  velocity  with  which  it  should  be  pro- 
jected ? 

The  formula  is 

log.  M=log.  0-Rog.  m  ;  (276) 

The  calculation  of  the  preceding  example  gives 

log.  ro=         .         .         .    .     .,     .         .         V"     0.0598930 
log.  (v=1000)        .  ...         3.0000000 

log.  (w=1147  feet)          .....         3.0598930 

(3).  A  24-pound  ball  being  projected  with  an  initial  velocity  of 
1200  feet  per  second,  —  required  the  time  it  will  take  to  pass 
through  the  first  900  feet  ? 

The  formula  is 


;  (278) 

we  have,  as  before, 

log.  m=0.0598930, 
and 

ro=1.147, 
m—  1=0.147; 

log.  m—  1,     ......         -         9.1673173 

log.  c,  (from  subsidiary  table),  .         .         .         3.8147507 

ar.  co.  log.  (w=1200  feet,  )      .         .  6.9208188 

log.  (/=0."799,)     ......          9.9028868 

or  nearly  8-tenths  of  a  second. 

4.  The  elevation  of  the  axis  of  a  24-pound  gun,  when  pointed 
de  But  en  Blanc,  is  1°.20'  ;  with  what  initial  velocity,  should  the 
ball  be  projected  to  strike  an  object,  distant  2000  feet  ? 

The  formula  is, 


log.  (ar—  2000,)  .....          3.3130300 

2 


log.  ar1,  ...    6.6160600 

ar.  co.  log.  c,  .        6.1852493 


2.8013093 


Book  IV^  PROJECTILES.  357 


PROJECTILES. 

X2 

—=  632.86  Brought  forward, 

#=2000.00 


—  +z  =2632.86  log.     .    .•   .;>  .    3.4204278 

log.  16,     .......    1.2041200 

ar.  co.  tan.  «',     .  1.6331055 


2)6.2576533 

log.  (tt=1345.32)  3.1288266 

(5).  At  what  angle  of  elevation  should  a  24-pound  gun  be  fired, 
in  order  to  strike  amark  at  a  distance  of26S2jeet,  the  initial  velo- 
city being  1500  feet  per  second  ? 
The  formula  is, 

y+*  )  (282) 

log.  (#=2682  feet,)         .          .         .          .    '     .         3.4284588 

2 


log.  x*        '.'V^  .....      >    .  6.8569176 

ar.  co.  log.  c,  ......  6.1852493 

a? 

log.  (—=1102)         V.V      ....  3.0421679 


#=2682 


—  \-x  =  3784  log  ......         3.5779511 

c 

log.  16,          -   .  •        1.2041200 

log.  u2=3.  1760913  X  2  ar.  co.        •    •    3.6478174 

tan.  (t=l°.32'.29/',)    .....    8.4298885 

(6)   What  is  the  range  de  But  en  Blanc,  of  a  24-pound  gun, 
whose  natural  elevation  is  1°.20'.,  and  initial  velocity  1400  feet  per 
second  1 
The  formula  is, 

tan.  i 


log.  (w=1400)      ......  3.1461280 

2 


log.  w3  Carried  over,  6.2922560 

33 


MOTIOXOF  [Book  IV. 

log.  wa                                   Brought  over,  6.2922560 

tan    (i  =  l°.20)  8.3668945 

ar.  co.  log.  16 8.7958890 

ar.  co.  log.  c 6.1852493 


log.  0.436797        .  .  9.6402798 

add  0.25 


log.  0.686797        »      '  »"  ~      *         •     .•-...?-'  9.8368285 

which  divided  by  2,  gives        ....          9.9684142 

whose  number  is  0.931996 
deduct  0.500000 


log.      .         .         0.431996  9.6354798 

log.  c          j  .v. •(-•*-       .      ="{-.tf(^'.'V-"'..       V         3.8147507 
log.  2819.88       \.  '      .'      .'      .         .     ^:*'       3.4502305 

The  distance  de  but  en  Wane  is  therefore  nearly  2820  feet. 

The  foregoing  tables  are  not  applicable  to  the  case  of  projectiles, 
fired  at  angles  exceeding  four  or  five  degrees,  in  consequence  of 
the  error  which  arises  from  taking  the  horizontal  distance  as  equal 
to  the  actual  path,  and  from  considering  the  tangent  of  the  angle 

i  ft/** 
of  elevation  to  be  equal  to .     Both  of  which  assumptions  are 

obviously  far  different  from  the  truth,  when  the  path  acquires 
any  sensible  curvature.  The  approximations  that  have  been 
made  to  the  determination  of  the  problem  of  the  motion  of 
shells,  and  other  projectiles,  fired  at  great  angles  of  elevation, 
may  be  seen  in  Hutton's  tracts.  In  actual  service,  tables  are 
used,  calculated  for  every  particular  species  of  ordnance;  and 
hence,  it  is  unnecessary  to  attempt  giving  any  general  rules. 

263.  The  value  which  we  have  taken  for  the  co-efficient  of  v3, 
in  the  formulae,  is  that  which  corresponds  to  initial  velocities  of 
12  to  1500  per  second.  Greater  velocities  may  be  given,  as 
has  been  stated,  to  military  projectiles  ;  say  upwards  of  2000  feet 
per  second.  To  give  these  velocities,  is,  however,  attended 
with  a  much  increased  expenditure  of  gunpowder ;  for  the  resist- 
ances increase,  afterwards,  in  much  higher  ratio.  The  cause  of 
this  is,  that  the  projectile,  in  passing  through  the  atmosphere, 
forms  a  wave  of  condensed  air  in  its  front,  while  the  air  is  rare- 
fied behind  it ;  it  is  hence  resisted  by  a  medium  of  greater  den- 
sity, while  it  derives  little  or  no  support  from  behind.  At  a  ve- 


Book  IP.]  PROJECTILES. 

locity  of  1342  feet,  the  air  will  no  longer  follow  it,  and  a 
vacu^wn  will  be  left  behind  it :  hence,  any  initial  velocity,  however 
gprat,  will  be  speedily  reduced  to  that  limit,  and  will  require  to 
j>roduce  it  a  greater  charge  of  gunpowder  than  would  be  con- 
sistent with  the  formula  (267) 


A  velocity  of  1300  feet  per  second  is  given  by  a  charge  of  one 
third  of  the  weight  of  the  ball  j  and  this  is  the  greatest  charge 
that  ought  to  be  admitted  in  service,  except  in  one  particular 
case,  to  which  we  shall  hereafter  refer.  This  is  the  regulation 
charge  for  cannon  in  the  British  and  American  navies  ;  and  al- 
though the  velocity  it  gives  is  often  greater  than  is  advisable,  for 
reasons  that  will  be  presently  stated,  still  it  would  be  inexpe- 
dient to  reduce  it  farther,  in  consequence  of  the  injury  that 
gunpowder  sustains  from  the  moist  air,  in  contact  with  the 
ocean.  In  the  land  service,  on  the  other  hand,  where  this  de- 
terioration does  not  take  place  to  such  an  extent,  the  service 
charge,  except  in  the  case  that  has  been  referred  to,  need  never 
exceed  |th  or  £th  of  the  weight  of  the  ball. 

264.  As  balls  for  the  larger  species  of  ordnance  are  made  of  a 
hard  material,  cast-iron,  they  cannot  fit  the  bore  of  the  piece  ex- 
actly, without  endangering  its  bursting.  Hence  there  is  a  differ- 
ence between  the  calibre  of  the  piece,  and  the  diameter  of  the 
ball,  which  is  called  the  windage.  The  more  perfect  the  work- 
manship of  the  bore,  and  the  more  accurately  the  balls  are  cast, 
the  less  may  be  the  windage.  It  has,  therefore,  been  considerably 
diminished  with  the  improvement  of  the  mechanic  arts.  'In  dif- 
ferent services,  the  practice,  in  this  respect,  has  been  different. 
In  some,  the  windage  has  been  made  to  bear  a  certain  proportion 
to  the  diameter  of  the  ball  ;  in  others,  it  is  a  constant  quantity  in 
all  cavities.  Were  the  principal  dangers  to  be  apprehended,  a  want 
of  sphericity  in  the  balls,  or  of  regularity  in  the  bore,  the  former 
would  be  the  correct  method  ;  but  in  the  present  state  of  the  arts 
in  our  country,  the  chief  risk  will  arise  from  an  increase  in  the 
diameter  of  the  ball  by  oxidation,  and  this  may  be  met  by  a  wind- 
age constant  in  all  calibres.  Sir  Howard  Douglas  has  proposed 
that  it  be  reduced  in  the  British  naval  service  to  little  more  than 
ith  of  an  inch,  from  0.33  in.  in  the  42  pound  gun,  and  0.22  in.  in 
the  12.  With  this  reduced  windage,  he  infers  that  an  equal  ve- 
locity may  be  given  with  fths  of  the  powder  now  used.  That 
an  increased  velocity  will  be  a  consequence  of  the  diminution  of 
the  windage  will  be  obvious,  when  it  is  considered  that  the  ball 
moves  from  rest  to  a  velocity  that  never  exceeds  fths  of  that  with 
which  the  elastic  fluid  tends  to  expand  itself.  Hence  a  portion 
of  the  gas  will  escape  without  acting ;  and  this  will  bo  deter- 


260  MOTION  OP  [Book  IV. 

mined  by  the  size  of  the  eccentric  ring,  formed  hy  the  circular 
sections  of  the  bore  and  the  ball. 

265.  In  consequence  of  the  windage,  the  ball  will  rest  on  tha 
bottom  of  the  bore,  or  will  strike  against  it  as  it  passes  out,  even, 
when  supported  by  a  wooden  seat  or  bottom,  in  such  manner 
that  its  axis  coincides  with  the  axis  of  the  piece.  In  either  of 
these  cases,  and  one  or  other  must  occur,  the  ball  will  acquire  a 
rotary  motion  around  an  axis  that  is  not  parallel  to  the  axis  of 
the  piece.  It  will  thus  happen,  as  will  be  easily  seen  from  §  241, 
that  the  ball  will  deviate,  not  only  from  a  parabolic  path,  but 
from  a  plane  curve;  that  it  will,  according  to  the  direction  of  the 
axis  around  which  the  rotary  motion  takes  place,  be  deflected  to 
the  right  or  to  the  left  of  the  vertical  plane,  passing  through  the 
axis  of  the  piece ;  and  that  it  may  rise  above,  or  fall  below  the  curve 
that  would  be  described  by  a  body  not  revolving.  In  striking 
against  the  bore,  it  may  be  reflected  and  thrown  to  the  opposite 
side ;  and  this  may  occur  more  than  once  before  it  leaves  the 
mouth  of  the  piece;  this  will  cause  a  change  in  the  initial  di- 
rection, and  concur  in  giving  a  rotary  motion,  by  the  combina- 
tion of  which,  very  considerable  deviations  may  occur.  These 
deviations  may  also  be  affected  by  irregularities,  the  ball  passing 
from  one  side  to  the  other  of  the  same  vertical  plane.  Such  de- 
viations may  arise  from  cavities  on  the  surface  of  the  balls,  or 
from  their  not  being  homogeneous  within. 

Inequalities  in  the  bore  may  also  contribute  to  cause  deviations  ; 
and  hence,  particular  guns  will  always  throw  their  shot  to  the 
right,  and  others  to  the  left  of  the  line  of  sight. 

The  same  weight  even  of  the  same  parcel  of  gunpowder,  will 
often  produce  different  initial  velocities  ;  and  different  parcels 
often  differ  in  strength.  From  all  these  circumstances,  the  prac- 
tice of  gunnery  is  attended  with  a  great  deal  of  uncertainty  ;  and 
even  the  best  theory  is  no  more  than  a  guide,  and  does  not  give 
results  that  are  to  be  implicitly  relied  upon. 

The  deviations  from  the  vertical  plane,  of  which  we  have  just 
spoken,  will  be  enhanced  by  increasing  the  velocity  ;  and  hence 
it  is,  that  charges  less  than  those  generally  employed,  will  be 
most  advantageous  in  many  cases.  For  if  a  ball,  after  passing 
through  a  given  space,  retains  sufficient  velocity  to  do  the  injury 
it  is  intended  to  effect,  it  will  strike  the  mark  with  more  cer- 
tainty, if  discharged  with  a  less  initial  velocity;  and  the  required 
range  will  be  better  attained  by  an  increased  elevation,  than  by 
an  increased  charge.  In  all  cases,  greater  uncertainty  in  the  at- 
tainment of  the  object  aimed  at,  is  produced  by  the  use  of  too 
great  a  charge  of  gunpowder. 

266.  The  deviation  of  projectiles  from  the  vertical  plane,  might 
be  obviated  by  giving  them  a  rotary  motion  around  an  axis  co- 


Book  IV.]  PROJECTILES.  261 

inciding  with  the  axis  of  the  piecp.  This  has,  however,  been 
found  impracticable  in  the  larger  species  of  ordnance.  In  small 
arms,  it  may  be  effected  by  what  is  called  rifling  the  barrel.  This 
consists  in  cutting  in  the  metal  that  surrounds  the  bore,  shallow 
grooves,  extending  from  the  muzzle  to  the  lodgment  of  the  ball; 
these  are  spiral,  making  about  one  revolution,  more  or  less,  ac- 
cording to  the  length  of  the  barrel,  around  the  bore.  The  ball 
being  of  a  soft  metal,  lead,  is  cast  a  little  larger  than  the  bore, 
and  is,  in  loading,  forced  in  ;  it  is,  therefore,  cut  by  the  hard 
sides  of  the  bore  into  a  form  adapting  itself  to  the  spiral  grooves 
or  rifles  ;  and  when  the  piece  is  discharged,  it  derives  from  them 
a  rotary  motion  around  the  axis  of  the  piece.  As  there  will  be 
no  windage,  a  less  proportionate  Charge  will  give  an  appropriate 
velocity  than  in  any  other  species  of  ordnance,  and  the  aim  is  far 
more  sure  and  certain. 

267.  It  has  been  seen  that  the  initial  velocity,  although  in- 
creased by  increasing  the  length  of  the  piecs^  increases  in  a  much 
less  ratio,  or  with  a  power  of  the  length  be^veen  \^s  square  and 
cube  root,  which  may  be  represented  by  f . 

It  has  also  been  seen  that  great  velocities  are  n^  attended  with 
proportionate  ranges,  and  cause  uncertainty  in  the-,im>  \^  may 
hence  be  concluded  that  neither  very  great  lengths  >j  the  bore, 
nor  large  charges  of  gunpowder,  are  ever  necessary.  Vith  small 
charges,  the  metal  of  the  piece  is  less  strained  than  w%  large 
and  thus  not  only  may  the  length,  but  the-  thickness  of  th*  piece 
be  reduced.  The  results  of  the  experiments  of  Robins  ana.  Hut- 
ton,  have  led  to  the  lessening  of  the  size  and  weight  of  mo^t  of 
the  pieces  of  ordnance.  A  great  and  sudden  improvement  w\s, 
in  consequence,  made  in  the  artillery  services  of  Europe,  about 
the  commencement  of  the  wars  of  the  French  revolution.  No 
field-piece  has  now  a  bore  of  more  than  18  calibres  in  length, 
which  is,  or  was,  lately,  the  regulation  in  the  French  service.  In 
the  English  service,  the  regulation  length  is  fourteen  calibres, 
while  in  the  American,  during  the  late  war,  it  was  reduced  to 
twelve,  and  the  pieces  weighed  no  more  than  Icwt.  to  each  pound 
of  ball.  These  were  found  to  be  sufficient  for  all  purposes  of  the 
service.  An  unwise  policy  has  lately  led  to  the  alteration  of  the 
model,  by  giving  the  bore  the  proportions  of  the  French  pieces, 
yet  without  increasing  the  weight;  it  has,  however,  been  found, 
that  pieces  of  the  new  model,  even  after  standing  proof,  have 
burst  in  the  schools  of  practice. 

In  cannon  other  than  field  pieces,  a  reduction  of  the  length  is 
not  always  practicable  :  thus  in  battering  guns,  a  certain  length 
in  front  of  the  trunnions  is  absolutely  necessary,  for  they  are, 
generally  speaking,  fired  from  embrasures  of  earth,  which  would 
be  injured  by  the  gas  expanding  in  every  direction  from  the 


MOTION  or  [Book  IV, 

mouth  of  a  short  gun.  The  battering  guns  of  the  French  service 
are,  therefore,  made  of  the  form  of  two  frusta  of  cones,  united  at 
the  trunnions;  that  nearest  the  breech  diminishing  more  rapidly 
than  that  towards  the  muzzle.  We  shall  see  that  this  form,  al- 
though well  adapted  for  this  object,  is  not  as  strong  as  one  formed 
on  another  principle.  The  three  calibres  of  French  battering 
guns  have  all,  for  the  same  reason,  equal  lengths.  Navy  guns 
should  also  project  to  a  certain  distance  beyond  the  side  of  the 
vessel  ;  and  the  same  reasons  apply  to  garrison  as  to  battering 
guns.  It  is,  besides,  convenient  to  have  them  of  the  same  model, 
that  the  same  pieces  may  be  used  for  either  purpose.  In  the 
American  service,  it  may  be  here  stated,  instead  of  the  three  ca- 
libres used  by  the  French  for  battering  guns,  the  24,  16,  and  12 
pounders,  there  is  but  one.  tne  18  pounder. 

In  field  pieces,  there  «  no  objection  to  shortening  the  smaller 
pieces,  and  hence  this'1335  of  guns,  of  each  different  nation,  have 
bores  of  a  constant  D»m°er  of  calibres  in  length. 

Cannon  were  fomerly  made  of  an  alloy  of  copper  and  tin, 
which  is  generaiy>  but  improperly,  called  brass.  It  had  the  ad- 
vantage of  a  g»°at  degree  of  tenacity,  but  was  objectionable  from 
its  great  den^Yt  and  high  cost.  It  was  also  liable,  in  rapid  ser- 
vice, to  softin  a°d  bend.  Now  that  the  charge  has  been  reduced, 
cast-iron  although  less  tenacious  than  the  brass,  has  been  sub- 
stituted f°r  it*  m  tne  ship,  garrison,  and  battering  guns  of 
Europ^m  nations.  But  it  has  not  been  found  practicable, 
gene-aMy  speaking,  to  use  cast-iron  in  the  lighter  species  of 
ordnance,  (field  pieces)  ;  for  although  it  is  a  general  rule  that 
snail  vessels,  of  similar  dimensions  and  material,  will  bear  a 
greater  strain  than  large  ones,  still  there  is  a  circumstance  in  the 
casting  of  iron  that  more  than  counteracts  this.  It  is  found  that 
articles  made  from  the  same  cast-iron,  and  drawn  from  the  same 
charge  of  a  furnace,  will  be  w.eaker  in  proportion  as  they  are  more 
rapidly  cooled  ;  and,  therefore,  the  small  masses  of  field  pieces,  cool- 
ing most  rapidly,  are  weaker  than  the  guns  of  other  descriptions. 
But  the  iron  made  from  the  ores  of  Sweden  and  the  United  States, 
with  charcoal,  is  of  such  excellent  quality,  that  field  pieces  may 
be  safely  made  of  it,  weighing  even  less  than  brass  guns  of  equal 
calibre  and  length. 

268  In  respect  to  the  shape  of  cannon,  it  will  be  at  once  seen 
that  the  part  in  which  the  charge  of  powder  is  situated,  must  bear 
the  greatest  strain  ;  and  it  seems  probable,  although  it  would  be 
difficult  to  reduce  it  to  the  test  of  mathematical  analysis,  that  the 
greatest  effort  is  exerted  by  the  expanding  gas,  at  the  point  where 
the  ball  is  lodged.  In  all  the  cases  of  burst  guns  that  we  have 
examined,  th^breech  was  entire,  a  fact  that  seems  to  corroborate 


Book  IV.]  PROJECTILES.  263 

this  inference.  Rumford  has,  therefore,  proposed  to  make  the 
thickness  of  metal  greatest  at  this  point,  and  has  planned  a  gun 
of  beautiful  proportions,  swelling  in  a  curve  from  the  breech  to 
the  point  assumed  for  the  lodgment  of  the  ball,  and  again  con- 
tracting in  a  curve  to  the  projection  of  the  muzzle.  It  is,  how- 
ever, impossible  to  assign  the  exact  point  at  which  the  ball  will 
ba  lodged,  in  consequence  of  the  difference  in  the  space  occupied 
by  different  charges  of  gunpowder.  Hence  the  form  given  to 
the  American  navy  32-pounder,  and  battering  18,  is  to  be  prefer- 
red ;  this  is  cylindric  from  the  base  ring  to  the  trunnions,  and 
conical  thence  to  the  swell  of  the  muzzle.  Our  navy  42-pounder, 
which  is  formed  on  the  principles  of  the  French  battering  gun, 
having  a  great  weight  in  the  breech,  and  being  comparatively 
thin  at  the  lodgment  of  the  ball,  is  much  weaker  under  an  equal 
weight,  than  if  it  were  formed  upon  the  same  principles  as  the  32- 
pounder. 

269.  The  next  circumstance  to  be  considered,  is  the  manner 
in  which  military  projectiles  penetrate  the  obstacles  they  encoun- 
ter. The  resistance  of  any  homogeneous  body  may  be  considered 
as  a  uniform  retarding  force  :  hence,  if  we  call  this  force/,  the  ve- 
locity with  which  the  ball  strikes  the  obstacle  v,  and  the  space  to 
which  it  penetrates  before  it  loses  its  whole  motion  s,  we  have 
from 


The  penetration  of  bodies  of  the  same  size,  figure,  and  weight, 
will,  therefore,  be  proportioned  directly  to  the  squares  of  their 
velocities,  and  inversely  to  the  strength  of  the  substance  into 
which  they  enter. 

If  the  bodies  be  balls  of  different  diameters  and  densities,  their 
own  moving  force,  (their  velocities  being  equal,)  will  be  propor- 
tioned to  their  densities,  and  the  cubes  of  their  diameters  ;  for  it 
is  as  their  masses  ;  the  body  into  which  they  enter,  will  resist 
with  a  force  proportioned  to  the  section  of  the  ball,  or  to  the  square 
of  its  diameter  ;  and,  therefore,  the  depth  penetrated,  will  be  as 
the  quotient  of  these  quantities,  or  directly  as  the  density  and  the 
diameter  of  the  ball. 

Introducing  the  cases  of  different  velocities  and  different  kinds 
of  obstacles,  we  have  for  the  general  rule  :  that  balls  penetrate 
into  obstacles  to  depths  that  are  proportioned  directly  to  their 
densities,  their  diameters,  and  the  squares  of  their  velocities; 
and  inversely,  to  the  resisting  force  of  the  obstacle. 

If  the  same  ball  strike  against  the  same  obstacle,  with  different 
velocities,  the  depths  are  proportioned,  as  has  been  shown,  to  the 
squares  of  the  velocities.  The  same  rule  will  apply  to  the  depth 


MOTION  OF  [Book  ir. 

that  remains  to  be  penetrated  after  a  portion  of  the  velocity  has 
been  lost,  and  the  remaining  velocities  will  be  proportioned  to  the 
square  roots  of  the  remaining  depths.  Let  us  suppose  the  whole 
depth  to  be  divided  into  units  of  length  ;  the  ratio  between  the 
original  velocity  and  that  remaining  after  penetrating  any  number 
of  units,  n  will  be 

>/(•-*) 

V/5        ' 

and  the  velocity  los.t  will  be 

Vs—  V(s—  n) 

-77-       '*• 

The  velocities  lost  in  passing  through  the  several  units  of  space, 
will  form  a  series  of  the  following  terms  : 


that  this  series  is  one  that  rapidly  in- 


And it  will  be  at  once  seen 


-  obstacle,  is  much  greate 


It  will  therefore  happen  that  balls  passing  with  great  veloci- 
ties through  thin  obstacles,  may  do  so  without  communicating 
any  perceptible  motion.  When  balls  pass  with  great  velocities 
through  a  substance  that  is  elastic,  the  hole  is  smooth,  and  is  often 
,of  less  diameter  than  the  ball  ;  but  if  the  velocity  be  small,  the 
obstacle  will  be  torn  and  rent. 

The  prodigious  effect  of  modern  military  engines  in  destroy- 
ing the  walls  of  fortresses,  depends  upon  the  principle  that  the 
penetration  is  proportioned  to  the  squares  of  the  velocities.  The 
actual  quantity  of  motion  in  the  battering  ram  of  the  ancients, 
might  be  made  equal  to  that  of  a  24-pound  ball  ;  but  the  former 
produced  no  other  effect  than  that  of  agitation,  by  which,  however, 
a  wall  might  be  destroyed,  after  a  long  series  of  blows.  When 
cannon  are  used  for  destroying  walls,  or  as  it  is  technically  called, 
battering  in  breach,  the  shot  of  all  the  guns  in  the  battery  are  first 
directed  so  as  to  cut,  by  their  penetration,  a  horizontal  groove  in 
the  lowest  part  of  the  wall  that  can  be  seen  from  their  position. 
They  are  next  directed  so  as  to  cut  two  vertical  grooves  at  each 
end  of  the  horizontal  one  ;  and  thus  a  quadrangular  portion  is  se- 
parated from  the  rest  of  the  wall.  The  cannon  are  then  fired  in 
vollies  against  this  portion,  until  the  agitation  it  receives  from 
the  shock  is  sufficient  to  cause  it  to  fall.  In  the  latter  part  of  the 
operation,  cannon  have  no  advantage  over  the  battering  ram,  but 


PROJECTILES.  265 

the  former  part  of  it,  by  which  it  is  in  fact  rendered  most  efficient, 
could  not  be  performed  by  the  ram. 

The  effect  being  proportioned  to  the  square  of  the  velocity, 
this  is  the  case  to  which  reference  was  made  in  §  261,  where  the 
charge  may  exceed  with  advantage  £d  of  the  weight  of  the  ball. 
It  is  usually  made  fds.  The  batteries  in  breach  are  also  estab- 
lished, as  near  as  possible,  to  the  wall  to  be  destroyed,  in  order 
that  the  velocity  may  be  but  little  diminished  by  the  air's  resist- 
ance. 

In  other  cases,  great  velocities  may  be  injurious  :  thus,  if  a  ball 
pass  through  the  obstacle,  it  will  communicate  less  motion  than 
if  it  lodge  ;  and  the  greater  the  velocity,  the  less  will  be  the  in- 
jury done.  For  this  reason,  in  close  naval  engagements,  great 
velocities  are  less  destructive  of  the  enemy's  vessel  than  smaller 
ones. 

Upon  this  principle  is  founded  the  introduction  of  the  car- 
ronade  into  the  naval  service.  This  species  of  ordnance  is  short ; 
and  being  loaded  with  no  more  gunpowder  than  |th  of  the  weight 
of  the  ball,  may  ha^e  but  little  thickness  of  metal.  It  may, 
therefore,  be  used  in  the  place  of  long  guns  of  much  smaller  ca- 
libres, while  the  effect  of  its  projectiles,  in  close  action,  is  greater 
than  that  of  a  ball  of  equal  weight  from  a  long  gun. 

270.  When  projectiles  strike  against  a  hard  substance,  in  an 
oblique  direction,  they  are  reflected  according  to  laws  that  will 
be  hereafter  examined.  So,  also,  on  the  surface  of  water,  balls 
impinging  at  small  angles,  rise  again,  and  perform  a  second 
curved  path  ;  and  this  may  be  repeated  several  times.  The  resist- 
ance of  the  water  will  be  proportioned  to  that  component  of  the 
moving  force  of  the  ball,  whose  direction  i*  a  normal  to  the  sur- 
face of  the  water :  when  this  becomes  less  than  the  gravity  of 
the  ball,  it  will  no  longer  rise.  This  motion,  in  successive  bounds 
over  the  surface  of  ground  or  water,  is  called  the  ricochet.  It  has 
been  applied  to  great  advantage  in  the  attack  of  fortified  places  ; 
and  gives  to  guns  placed  upon  the  shore,  in  proper  positions,  great 
advantages  over  those  opposed  to  them  in  ships.  In  the  attack 
of  fortified  places,  the  first  or  more  distant  batteries  are  no  longer 
placed  ia^front  of  the  part  to  be  attacked,  but  in  the  prolongation 
of  its  faces,  and  opposite  to  the  returning  sides  of  the  fortress. 
The  guns  in  these  batteries  are  fired  at  elevations  of  4°  or  5°, 
with  charges  of  gunpowder  that  enable  the  balls  in  the  descend- 
ing part  of  their  path,  just  to  raze  the  opposing  parapet:  they, 
therefore,  bound  along,  parallel  to  the  direction  of  the  front  to  be 
attacked,  dismount  the  guns,  and  destroy  the  defenders. 

Under  the  fire  of  these  ricochet  batteries,  approaches  are  made 
to  points  sufficiently  near  for  the  erection  of  batteries  in  breach  ; 

34 


' 

266'  MOTION  OF,  &LC.  [Book  I  y. 

by  these  the  walls  are  destroyed.  It  is  in  consequence  of  this  me- 
thod, which  was  invented  by  Vauban,  that  the  means  of  the  at- 
tack of  fortresses  have  become  superior  to  those  of  defence,  and 
that  the  time  of  the  resistance  of  a  fortress  can  be  calculated  with 
almost  mathematical  precision. 

When  balls  are  fired  from  shipping,  and  strike  the  water,  they 
never  rise  in  any  of  their  bounds  as  high  as  the  point  whence 
they  are  projected.  Hence,  in  firing  from  a  ship  at  an  object  on 
the  land  that  is  higher  than  the  deck  on  which  the  gun  is  placed, 
there  is  no  other  chance  of  striking  it  but  by  a  direct  aim  ;  while, 
if  a  ball  be  fired  from  the  land,  it  will  strike  the  vessel,  if  it  be 
within  the  limit  of  its  recochets  ;  and  if*the  height  of  the  battery 
be  not  so  great  as  to  permit  the  balls  to  rise  above  the  vessel.  It 
thus  happens  that  two  or  three  heavy  guns,  placed  upon  the  land 
in  a  proper  position,  are  more  than  a  match  for  the  heaviest  ship, 
while  towers,  and  walls  with  embrasures,  may  be  destroyed  by 
the  superiority  in  the  quantity  of  guns  that  can  be  arranged,  in  a 
given  space,  on  shipboard. 


Book  ]VJ\  THEORY  OF  THE  PENDULUM.  267 


CHAPTER  IV. 

THEORY  or  THE  PENDULUM. 

271.  If  a  gravitating  body  be  suspended  by  a  string  or  rod  from 
a  fixed  point,  it  will  hang  in  a  vertical  position  ;  but  if  it  be  raised 
from  that  position  laterally,  the  string  or  rod  remaining  inflex- 
ible, and  then  permitted  to  escape,  it  will,  under  the  joint  action 
of  gravity,  and  the  tension  of  the  string  or  rod,  descend  in  an 
arc  of  a  circle  of  which  the  point  of  suspension  will  be  the  centre. 
When  it  reaches  the  vertical  position,  it  will,  §  58  have  acquired 
a  velocity  equal  to  that  acquired  by  falling  vertically  through  the 
versed  sine  of  the  arc,  and  which  would  tend  to  carry  it  forwards 
with  uniform  velocity  in  a  horizontal  direction.     The  tension  of 
the  string  will  cause  it  to  continue  to  move  in  the  circle  of  which 
the  point  of  suspension  is  the  centre ;  it  will,  therefore,  after 
passing  the  vertical  line,  rise  in  a  circular  arc,  until  its  whole 
velocity  be  destroyed,  which,   if  no  other  force  but  gravity  act, 
will  be,  when  it  reaches  a  height  on  the  opposite  side  of  the  ver- 
tical, equal  to  that  whence  it  at  first  fell.      On  reaching  this  point, 
it  will  again  descend,  and  passing  the  vertical,  rise  to  the  point 
whencelt  originally  set  out.     From  this  it  will  a  second  time 
descend,  and  would  thus  continue  to  vibrate  for  ever.     But  it 
will  be  seen  that  two  forces  act  to  retard  the  motion,  namely,  the 
resistance  of  the  atmosphere,   and   friction  around  the  point  of 
suspension.     By  virtue  of  these,  the  circular  arcs  described  are 
gradually  lessened,  until  a  state  of  rest  be  reached;  the  suspend- 
ing string  or  rod  becoming  stationary  in  a  vertical  position. 

A  body  thus  suspended  and  caused  to  vibrate,  is  called  a  Pen- 
dulum. 

The  nassage  from  its  highest  position  on  one  side  of  the  ver- 
tical, until  it  reach  the  greatest  height  on  the  opposite  i  ide,  u 
called  an  Oscillation. 

272.  In  order  to  study  the  theory  of  this  species  of  motion, 
we  imagine  to  ourselves'a  gravitating  point,  suspended  by  meai 
of  an  inflexible  line,  devoid  of  gravity.     The  case  then  becomes 
that  of  &  60,  a  gravitating  point,  compelled  to  move  upon  a  cir 
cular  surface,  by  an  accelerating  force  whose  direction  is  always 
parallel  to  itself. 


268  THEORY  OF  [JBook  IV. 

The  time  t,  of  descent  in  a  small  circular  arc,  of  which  a  is  the 
radius,  and  h  the  versed  sine,  is  by  (84) 
<jf     a  h 


hence  the  whole  time  of  an  oscillation,  T,  becomes,  substituting 
/  the  length  of  the  pendulum  for  a, 

T=W1X(1+-|;  (285) 

O 

and  finally,  in  evanescent  arcs, 

T=W-  ; 

o 

whence  we  obtain  for  the  values  of  /  and  g4, 


and  when  T  =  l", 

g=W.  (288) 

To  compare  the  time  of  the  oscillation  of  a  pendulum  in  a  very 

small  arc,  with  that  of  the  descent  of  a  heavy  body  :  Let  the  dis- 

tance the  body  falls  be  half  the  length  of  the  pendulum,  and  we 

have  by  (61) 

'=4 

which  compared  with  (286)  gives 

T  :«::*:  1.  (289) 

When  the  respective  times  in  which  two  pendulums  vibrate,  or 

their  numbers  of  oscillations,  in  equal  times,  are  known,  and  the 

length  of  one  of  them  has  been  determined,  that  of  the  other  may 

be  calculated  — 

Let  /  and  I'  be  their  respective  lengths  ; 
T  and  T'  their  times  of  oscillating  ; 
N  and  N'  the  numbers  of  vibrations  in  equal  times,  then 
T  :  T'  :  :  N'  :  N  ; 

for  the  number  of  oscillations  is  inversely  as  the  times. 
We  have  for  the  value  of  the  times  (286) 


whence,  T  :  T'  ::>//:  ^/', 

and  N:  N':  :  Jl'  :  v'/; 

therefore,  when  the  respective  times  are  known,  and  I  is  given, 


Book  IV.}  THE  PENDULUM.  *C9 

and  when  N  and  N'  are  known,  and  I'  given, 

N"J' 
l=-jj»   .  (290) 

From  these  formulae  it  may  be  concluded  : 

(1.)  That,  the  force  of  gravity  remaining  constant,  the  times 
of  the  vibrations  of  pendulums  of  unequal  lengths  are  respectively 
as  the  square  roots  of  their  lengths,  and  inversely  as  the  square 
roots  of  the  numbers  of  their  vibrations  in  a  given  time. 

(2.)  That,  the  force  of  gravity  still  remaining  constant,  the 
lengths  of  different  pendulums  are  directly  as  the  squares  of  the 
times  of  their  oscillations,  and  inversely  as  the  squares  of  the 
numbers  of  their  oscillations  in  a  given  time. 

(3.)  That  the  time  of  the  oscillation  of  a  pendulum,  is  to  the 
time  that  a  heavy  body  would  fall  freely  by  the  force  of  gravity, 
through  half  its  length,  as  the  circumference  of  a  circle  is  to  its 
diameter. 

(4.)  In  different  positions,  the  intensity  of  the  force  of  gravity 
may  be  measured  by  the  length  of  the  pendulum  that  vibrates  in 
equal  times  at  the  different  places  ;  and  the  intensity  of  gravi- 
tation is  always  directly  proportioned  to  the  length  of  the  pendu- 
lum that  beats  seconds  at  the  place. 

273.  If  the  pendulum  vibrate  in  a  cycloid,  the  formula 


becomes  true,  §  59,  whatever  be  the  amplitude  of  the  arc  :  all 
the  above  propositions  are,  therefore,  absolutely  true  of  pendu- 
lums vibrating  in  cycloidal  arcs. 

It  has  been  attempted  to  make  pendulums  vibrate  in  cycloidal 
arcs,  upon  the  following  principles,  which  are  theoretically  true  ; 
although,  as  we  shall  hereafter  see,  they  are  not  susceptible  of 
being  applied  to  any  practical  purpose.  This  attempt  has  been 
founded  on  the  following  principles. 


rt- 


270 


THEORY  OF 


[Book  IV. 


It  is  a  property  of  the  cycloid,  that  its  evolute  is  an  equal  cy- 
cloid, placed  in  an  inverted  position.  Hence,  if  the  length  of 
the  pendulum  SP  be  bisected,  and  on  the  line  ADB,  drawn 


through  the  point  of  bisection,  perpendicular  to  SP,  a  cycloid  be 
constructed,  the  diameter  of  whose  generating  circle  is  DP;  and 
if  two  half  cycloids  be  constructed  tangents  to  the  lines  SD,  and 
ADB,  the  latter  will  be  the  evolutes  of  the  corresponding  semicy- 
cloids.  If  then  a  pendulum  be  suspended  from  S,  by  a  flexible  rgd, 
of  the  length  SP,  it  will,  when  moved  from  its  vertical  position  and 
permitted  to  oscillate,  apply  itself  to  the  two  curves  $A,  SB  ;  and 
it  is  evident,  that  during  these  oscillations,  it  will  describe  the 
curve  APB,  or  a  part  of  it. 

When  the  circular  arc  does  not  exceed  4  or  5  degrees,  the  re- 
lation of  the  time  of  oscillation  in  it  to  that  in  a  cycloidal  arc  is 
given  by  the  formulae  (2S5)  and  (2S6),  or  is  as 

l:l+i&:  (291) 

when  the  arc  exceeds  that  amount,  it  becomes  necessary  to  intro- 
duce more  terms  of  the  series  in  (84). 

From  this  series,  the  excess  of  the  times  of  oscillation  in  circular 
arcs  over  those  in  a  cycloid,  has  been  calculated,  as  follows,  viz. 
In  an  arc  of  30°  on  each 

side  of  the  vertical          .         .         .         0.01675 
of  15°  ....          0.00426 

of  10°  ....         0.00190 

of    5°  .          .          .  0.00012 

of    21°  ....          0.00003 

So  thifc  when  the  circular  arc.  in  which  a  pendulum  vibrates, 
does  not  exceed  2^°  on  each  side  of  the  vertical,  or  5°  in  all,  the 


Book  /KJ  TIIJ:  PENDULUM.  271 

excess  in  the  length  of  the  time  of  its  oscillations  over  those  in  a 
cycloid  does  not  exceed  9"  per  day. 

274.  It  has  been  seen,  §  96,  that  the  intensity  of  gravity  va- 
ries as  we  proceed  from  the  surface  of  the  earth,  in  the  inverse 
ratio  of  the  squares  of  the  distances. 

If  we  call  the  radius  of  the  earth  R,  the  distance  above  the 
mean  surface  A,  the  force  of  the  gravity  at  the  surface  g,  and  at 
the  height  ft,  g',  we  have 


whence  we  obtain 

2ft 


(2ft      ft2  \ 
l  +  tt~+B2l     • 
It  R2/     5 

or,  neglecting  the  last  term, 

[~       2ft~l  2g'  h 


and 


In  the  same  manner  we  obtain  for  the  values  of  the  lengths  of 
the  pendulums,  at  the  surface,  and  at  the  height  ft, 


These  relations  are,  however,  only  true  in  the  case  impossi- 
ble in  practice,  that  the  pendulum  is  raised  through  the  air  with- 
out resting,  as  it  must,  upon  some  projecting  part  of  the  solid 
crust  of  the  earth.  In  this  event,  a  local  attraction  of  the  mass  of 
ground  on  which  it  rests  will  interfere  with  the  law. 

275.  The  length  of  the  pendulum  that  vibrates  seconds  at  any 
place,  is  proportioned,  as  has  been  stated,  to  the  force  of  gravity 
at  that  part  of  the  earth's  surface.  That  is  to  say,  to  the  differ- 
ence between  the  whole  force  of  gravity  and  the  centrifugal  force. 
We  may  hence  obtain  the  relation  between  the  lengths  of  pendu- 
lums in  different  latitudes. 

If  the  centrifugal  force  at  the  equator  be  called/,  the  latitude 
L,  the  force  of  gravity  at  the  pole,  or  the  absolute  measure  of  that 
force,  G,  the  relative  force  at  the  equator  g-, 

g-=G-/;    andG=ff+/5 

and  the  centrifugal  force  at  the  latitude  L,  will  be,  §  100, 
/cos.  3L. 


THEORY  OF 

Then  if  g'  be  the  apparent  force  of  gravity  at  the  latitude,  ] 

g-'=G-/cos.aL; 

substituting  the  value  of  G  from  the  above  equation, 
g/=g>+/-/co8. 2  L=g+/(l-cos. 2  L), 

£'=£+/ sin.  2L.  (2 

As  the  pendulums  of  the  different  places  have  the  same  r 
as  the  gravitating  forces,  we  have,  if  E  be  the  length  of  the  [ 
dulum  at  the  equator,  and  d  the  difference  between  the  length 
the  polar  and  equatorial  pendulums,  for  the  difference  betw 
the  length  m  of  the  pendulum  at  the  equator,  and  at  the  latil 

w=E+dsin.  2L;  (2 

and  for  the  length  n,  at  another  latitude  L', 

n=E  +  d  sin. a  L'.  (2 

If  m  and  n  be  given,  the  lengths  at  the  pole,  and  the  equ 
may  be  found ;  for  by  subtraction  of  the  foregoing  express 
we  obtain 

m— n= (E+d  sin. 2  L)— (E-fd  sin. 2  L')  =<2(sin. 2  L— sin. 2 
whence  we  obtain  for  the  value  of  d, 

m — n 

sin.  (L+L') .  sin.(L— L') ; 

and  d  being  given,  we  have  the  value  of  E  from  either  of  the 
muhE,  (295)  and  (296), 

E=m — dsin.  2L     I  ff 

E^TI— dsin.  2L'    ( 

The  ratio—-  will  be  the  same  as^,  which  expresses  the  din: 
E  # 

tion  of  gravity  from  the  pole  to  the  equator. 

When  this  ratio  is  given,  the  ellipticity  of  the  terrestrial  sphe 
may  be  determined  ;  for  it  has  been  shown,  by  writers  on  j 
sical  astronomy,  that  if  this  ratio  be  subtracted  from  5  of  the 
portion  of  the  centrifugal  force  at  the  equator,  to  the  appa 
force  of  gravity  there,  the  remainder  expresses  the  oblatene* 
flattening  at  the  poles. 

When  more  than  two  observations  are  to  be  combined,  it 
be  best  done  by  the  method  of  least  squares,  as  may  be  sec 
the  Mecanique  Celeste,  in  a  paper  of  Dr.  Adrain   in  the  An 
can  Philosophical  Transactions,  and  in  Sabine's  Experiment 
the  Pendulum. 

276.  The  Simple  Pendulum,  such  as  we  have  assumed  it 
the  purpose  of  investigating  the  theory,  cannot  exist  in  prac 
for  we  can  neither  make  use  of  a  body  so  small  as  to  be  consid 
as  a  single  gravitating  point,  nor  abstract  from  the  weight  oi 


Book  IV.  THE  PETfDULUM.  273 


rod  by  which  it  is  suspended.  Such  pendulums  as  can  be  actually 
constructed,  are  called  Compound  Pendulums. 

in  every  compound  pendulum,  there  is  necessarily  a  point,  in 
which  if  all  the  matter  were  collected,  a  simple  pendulum  will 
be  formed,  whose  oscillations  will  be  performed  in  the  same  time 
as  those  of  the  compound  pendulum.  This  point  is  called  the 
Centre  of  Oscillation.  Its  essential  property  is,  obviously,  that 
the  sum  of  the  moments  of  rotation  of  the  several  points  of  which 
it  is  made  up,  will  be  equal  to  the  moment  of  rotation  of  the  whole 
mass,  if  situated  at  the  centre  of  osciUation. 

Calling  the  distance  from  the  centre  of  oscillation  ar,  the  seve- 
ral points  of  which  the  body  is  made  up,  A,  B,  C,  &c.,  their  res- 
pective distances  from  the  centre  of  suspension,  a,  6,  c,  &c.,  we 
have  for  the  sum  of  their  moments  of  rotation,  by  §  241, 

(Aa2-l-B624-Cc2+&c.)  -  ; 

and  for  the  moment  of  rotation  of  the  whole  mass,  if  collected  in 
the  centre  of*  oscillation, 


(Aa+B6+Cc+&c.)  x  -  ; 
a 


whence  we  obtain 


Hence,  the  centre  of  oscillation  in  a  vibrating  body  is  identical 
with  the  centre  of  percussion  in  a  revolving  body  ;  and  the  rule 
for  finding  its  position  may  be  thus  expressed  in  words  :  Divide 
the  moment  of  inertia  of  the  body  by  its  moment  of  rotation;  the 
quotient  is  the  distance  of  the  centre  of  oscillation,  from  the  cen- 
tre of  suspension. 

If  the  points,  instead  of  being  united  by  a  line  devoid  of  weight, 
compose  a  solid  body,  the  same  proposition  will  be  true,  provided 
the  distances  be  measured  from  the  several  points,  perpendicularly 
to  the  horizontal  line  passing  through  the  point  of  suspension  ;  and 
if  the  body,  instead  of  being  suspended  by  a  point,  is  supported  on 
the  last  mentioned  line  as  an  axis,  the  propositions  are  still  true  in 
respect  to  it.  So,  also,  if  a  horizontal  line  be  drawn  through  the 
centre  of  oscillation,  every  point  in  it  will  be  at  an  equal  distance 
from  the  axis  of  suspension  ;  and  hence  any  point  in  the  former 
line,  which  may  be  called  the  axis  of  oscillation,  will  have  the 
properties  of  a  centre  of  oscillation. 

From  the  properties  of  the  centre  of  gravity,  the  quantity 

Aa+B6-fCc-h&c. 

is  equal  to  the  mass  multiplied  by  the  distance  of  the  centre  of 
gravity  of  the  body,  from  the  centre  of  suspension,  or  calling  the 

35 


274  THEORY    OF  [Book  IV. 

former,  M,  the  latter,  «-,  to  Mg.  And  if  we  call  the  moment  of 
inertia,  S,  the  formula,  (299),  becomes 

S 

-S£' 

and 

S=Mgx.  (300) 

If  the  pendulum  be  suspended  by  its  axis  of  oscillation,  that 
being  parallel  to  the  axis  of  suspension,  the  moment  of  rotation, 
S',  now  becomes,  (259), 

S'  =  S—  m(k'2—  #•)  , 
=  S—  m(k'+k)  (k'—k)  ; 

but  k'+k  is  the  distance  between  the  axes  of  suspension  and  os- 
cillation, or  =x,  therefore, 

S'=S—  mkx+mk'x; 
but,  by  (246), 


hence, 

S'=mkx  ; 

and  the  moment  of  rotation  is  mk  ;  hence,  the  distance,  #',  from 
the  axis  of  oscillation,  when  the  pendulum  is  suspended  by  it,  to 
the  line  that  becomes  the  axis  of  oscillation,  is 

•Ha?--  (301> 

277.  Thus  it  appears,  that  if  a  compound  pendulum  be  sus- 
pended by  its  axis  of  oscillation,  the  axis  of  suspension  becomes 
the  axis  of  oscillation,  or  that  the  centres  of  oscillation  and  suspen- 
sion are  convertible  points.  Therefore,  if  a  pendulum  be  suspended 
upon  an  axis  passing  through  its  centre  of  oscillation,  it  will  vi- 
brate in  the  same  time  as  when  suspended  from  its  ordinary  axis 
of  suspension  ;  and  conversely,  if  a  pendulum  be  found  to  vibrate 
in  equal  times  when  suspended  from  two  different  axes,  and  if  one 
of  these  be  called  the  axis  of  suspension,  the  other  becomes  the  axis 
of  oscillation  ;   and  the  distance  between  these  axes  is  the  length  of 
a  simple  pendulum,  whose  oscillations  would  be  isochronous,  with 
those  of  the  compound  pendulum. 

278.  When  the  length  of  a  pendulum  is  spoken  of,  we  do       j 
understand  its  physical  length,  or  distance  between  its  two  ex- 
treme ends  ;  but,  by  this  term  we  understand  the  distance  between 
its  centres  of  oscillation  and  suspension,  or  the  length  of  the  iden- 
tical simple  pendulum.     The  determination  of  the  position  of  the 
centre  of  oscillation  of  a  pendulum,  therefore,  frequently  becomes 
an  important  subject  of  inquiry  in  practice.  This  may  be  effected 
in  the  case  of  all  solids  formed  by  the  revolution  of  symmetric 
curves,  by  means  of  the  principle  in  §  276,  that  the  distance  be- 


Book  IF.]  THE  PENDULUM.  275 

tween  the  centres  of  suspension  and  oscillation,  may  be  found  by 
dividing  the  moment  of  inertia  by  the  moment  of  rotation,  both 
in  reference  to  the  axis  of  suspension. 

(1).  Thus  in  the  case  of  a  straight  line,  suspended  by  one  of 
its  extremities,  the  moment  of  inertia  is  ~,  (247),  the  moment 

of  rotation  ~  ;  hence,  we  have  for  the  distance  between  the  cen- 
tres of  oscillation  and  suspension, 

'=y.  (802) 

In  the  same  manner  we  may  obtain,  using  the  formulae,  (253)  and 
(254) : 

(2).  In  a  cylinder  suspended  by  the  middle  of  one  of  its  bases, 
a  being  the  length,  and,  6,  the  radius  of  the  base, 

2a      b 

~3~+2^'  (303) 

(3).  In  a  cone  suspended  by  the  vertex,  the  radius  of  whose 
base  is  6, 

4a     ba 

/=T+5^  <304) 

When  the  cone  is  a  right  cone, 

a=b, 
and 

l=a. 

In  which  case  the  centre  of  oscillation  is  in  the  middle  of  the  base. 
(4).  In  a  sphere  whose  radius  is  r,  and  which  is  suspended  from 
a  point  on  its  surface,  from  (256) 

l==~  .  (305) 

(5).  In  a  sphere  whose  radius  is  r,  and  which  is  suspended  by 
a  line  devoid  of  weight,  that  unites  a  point  in  its  surface  to  the 
centre  of  suspension  :  if  c  be  the  length  of  this  line,  we  have,  by 
using  the  principles  of  §  250, 

.  2r* 


279.  It  has  already  been  mentioned  that  pendulumsin  all  the  ca- 
ses in  which  they  can  be  employed,  are  resisted  by  the  friction  upon 
the  axis  of  suspension,  and  by  the  resistance  of  the  medium  in  which 
they  move,  namely,  the  atmosphere.  The  arcs  described,  therefore, 
gradually  become  less  and  less,  until  the  motion  ceases  altogether, 
and  the  pendulum  reaches  a  state  of  rest.  The  resistance  at  the 
axis  of  suspension  is  usually  diminished  to  the  lowest  practicable 


276  THEORY  OF,  &c.  [Book 

limit,  and  hence  it  becomes  unnecessary  to  apply  any  correction 
for  it,  except  so  far  as  its  effect  becomes  apparent  in  the  gradual 
lessening  of  the  are  of  oscillation. 

In  respect  to  the  resistance  of  the  air,  as  the  velocity  of  the 
pendulum  is  at  most  but  small,  that  part  of  it  which  varies  with 
the  square  of  the  velocity  may  be  neglected,  and  the  resistance 
may  be  considered  as  directly  proportional  to  the  velocity.  Were 
this  absolutely  true,  although  the  motion  would  be  retarded,  and 
each  oscillation  increased  in  duration,  the  time  of  the  performance 
of  each  would  be  equally  affected,  and  the  isochronism  in  arcs  of 
a  cycloid  would  not  be  altered,  while  in  circular  arcs  the  variation 
in  isochronism  would  be  allowed  for,  in  the  correction  for  the  ex- 
tent of  the  arc.  The  whole  retardation  would,  if  this  were  true, 
be  allowed  for  by  applying  a  correction  for  the  buoyancy  of  the 
air.  The  gravity  of  the  pendulum  is  diminished  by  its  falling 
through  that  fluid;  and  hence,  if  m  represent  the  absolute  density 
of  the  pendulum,  m',  the  density  of  the  air,  the  correction 
would  be 


m  —  m 


Such,  at  least,  is  the  only  correction  that  has  hitherto  been  em- 
ployed to  determine,  from  the  actual  oscillations  of  a  pendulum, 
what  would  occur  were  it  to  move  in  a  vacuum  or  space  void 
of  air.  Recent  experiments  by  Bessel,  and  by  Sabine,  have  shown 
that  this  correction  is  insufficient.  Bessel's  experiments  consisted 
in  making  pendulums  of  similar  figures,  but  of  unequal  densities, 
vibrate  in  air  ;  and  in  making  the  same  pendulums  vibrate  in  two 
different  media,  say  air  and  water.  From  these,  he  inferred  the 
difference  that  would  ensue,  were  the  pendulum  to  move  in  a  space 
void  of  air.  Sabine,  on  the  other  hand,  instituted  a  direct  com- 
parison between  the  oscillations  of  a  pendulum  in  air,  and  in  such 
a  vacuum  as  may  be  obtained  by  the  air-pump.  These  different 
modes  of  experimenting  lead  to  similar  conclusions,  namely,  that 
the  correction  necessary  to  be  applied  to  reduce  the  motion 
in  air,  to  that  in  a  vacuum,  is  in  all  cases  greater  than  is  re- 
presented by  the  formula  that  has  just  Keen  stated  ;  and  that  it 
depends  upon  the  figure  of  the  pendulum.  It  must,  therefore, 
be  ascertained  for  every  different  shape  that  is  given  to  the  pendu- 
lum, by  actual  experiment,  for  it  can  hardly  be  susceptible  of  re- 
duction to  any  precise  formula.  The  consequence  of  this  disco- 
very, upon  some  experiments  with  the  pendulum,  will  hereafter 
be  cited. 


Book  IF'.]  APPLICATIONS  OF  THE  PENDULUM.  877 


CHAPTER  V. 

APPLICATIONS  OP  THE  PENDULUM. 

280.  The  pendulum  may  be  applied  to  three  several  impor- 
tant purposes. 

(1.)  To  measure  portions  of  time,  or  to  subdivide  the  units 
we  derive  from  astronomic  phenomena,  into  smaller  and  equal 
portions. 

(2.)  To  determine  the  measure  of  the  force  of  gravity,  at 
different  places,  and  under  different  circumstances;  and  thus  to 
enable  us  to  infer  the  variation  in  the  apparent  intensity  that  is 
due  to  the  centrifugal  force;  and  the  variation  in  the  actual  in- 
tensity at  the  surface,  that  is  due  to  the  figure  of  the  earth.  Hence 
the  figure  of  the  earth  may  be  inferred. 

(3.)  The  pendulum  has  been  employed  as  a  standard  of  mea- 
sure. 

281.  The  pendulum  is  employed  as  a  measure  of  time,  upon 
the  principle  that  its  oscillations,  in  very  small  circular  arcs,  are 
isochronous,  provided  the  length  of  the  pendulum  do  not  vary, 
and  the  intensity  of  gravity,-  at  a  given  place,  remains  constant. 
The  latter,  we  have  every  reason  to  believe,  is  unchangeable  in 
the  same  part  of  the  earth's  surface,  and  it  will  be  seen  that  there 
are  methods  by  which  the  former  may  be  rendered  nearly  inva- 
riable also. 

When  pendulums  are  employed  as  a  measure  of  time,  they  are 
adapted  to  instruments  called  Clocks.  These  instruments  an- 
swer the  two-fold  purpose  : 

(1.)  Of  restoring  to  the  pendulum,  at  each  oscillation,  as  much 
force  as  it  loses,  in  consequence  of  the  resistance  of  the  air,  and 
the  friction  at  the  axis  of  suspension  ;  and 

(2.)  Of  registering  the  number  of  oscillations  performed  by 
the  pendulum.  To  accomplish  the  latter  object,  they  are  so  con- 
structed, that  by  simple  inspection,  the  interval  of  time,  or  num- 
ber of  oscillations  that  have  elapsed  since  the  clock  was  last  ob- 
served, can  be  at  once  determined.  The  rate  of  the  oscillations 
of  the  pendulum  can  be  changed  by  altering  its  length  ;  and 
thus  by  comparing  the  clock  with  astronomic  observations,  the 
day  may  be  divided  into  the  usual  number  of  conventional  parts. 
The  greater  number  of  clocks  have  pendulums  that  oscillate  once 
in  each  second  of  time  ;  and  when  we  speak  of  the  length  of  a 
pendulum  at  a  given  place,  we  mean  one  whose  beats  are  an  ex- 


APPLICATIONS    OF  [Book  IV. 


act  second,  or  jsho  part  of  a  mean  solar  day.  Some  clocks  have 
pendulums  that  beat  half  seconds,  and  others  again  oscillate  but 
once  in  two  seconds.  According  to  the  principles  in  §  271,  the 
pendulums  of  the  former  must  have  no  more  than  one  fourth  of 
the  length,  and  of  the  latter,  must  be  four  times  as  long  as  the 
second's  pendulum. 

It  was  at  one  time  attempted  to  make  the  pendulums,  used  as 
measures  of  time,  vibrate  in  arcs  of  a  cycloid.  This  has  been 
abandoned,  in  consequence  of  the  mechanical  difficulties  of  mak- 
ing the  cycloidal  cheeks,  to  which  it  should  adapt  itself:  of  the 
impossibility  of  obtaining  a  flexible  string  of  invariable  length  ; 
and  of  the  doubt  that  must  exist  in  pendulums  of  any  convenient 
form,  ifl  respect  to  the  position  of  tho  point,  the  centre  of  oscilla- 
tion, that  ought  to  describe  the  curve. 

282.  The  pendulum  of  a  clock  is  usually  composed  of  a  weight, 
or  bulb,  of  a  lenticular  form,  adapted  to  an  axis  of  suspension  by  a 
metallic  rod.  As  both  the  rod  and  the  bulb  are  liable  to  vary  in 
magnitude,  by  variations  in  temperature,  the  position  of  the  cen- 
tre of  oscillation,  and  consequently  the  length  of  the  pendulum, 
must  be  undergoing  perpetual  alterations.  Thus  the  law  of  iso- 
chronism  will  be  no  longer  true,  and  the  clock  will  not  move  at 
an  uniform  rate,  nor  mark  equal  divisions  of  time.  To  obviate 
this  defect,  various  modifications  of  the  original  simple  form  have 
been  contrived,  which  are  called  Compensation  Pendulums. 
Their  general  principle  is  identical,  and  consists  in  making  the 
pendulum  of  two  substances  that  expand  in  opposite  directions, 
in  such  a  manner  as  to  keep  the  centre  of  oscillation  at  a  constant 
distance  from  the  axis  of  suspension. 

Various  methods  have  been  proposed  and  employed  for  this 
purpose  : 

(1.)  The  rod  of  a  clock  pendulum  is  often  attached  to  a  short 
piece  of  flexible  metal  or  spring  ;  this  may  be  griped  by  two  plane 
surfaces,  pressed  against  it  by  screws  ;  and  its  effectual  length 
will  be  measured  from  the  lower  edge  of  these  surfaces,  provided 
the  pressure  be  sufficient  to  support  its  whole  weight.  Now  if  a 
bar  of  the  same  metal  with  that  which  forms  the  rod  of  the  pen- 
dulum, and  of  an  equal  length,  be  adapted  to  the  back  of  the 
clock-case  ;  and  if  it  be  so  applied  to  a  horizontal  arm,  attached 
to  the  spring  that  bears  the  pendulum,  that  it  shall  in  its  contrac- 
tions and  expansions  cause  the  spring  to  slide  between  the  surfaces 
on  which  it  rests  ;  every  change  in  the  length  of  the  pendulum 
rod,  under  the  influence  of  temperature,  would  be  exactly  coun- 
teracted. The  objection  that  applies  to  this  method  is,  that  if 
the  friction  of  the  surfaces  against  the  spring  be  sufficient  to  bear 
the  whole  weight  of  the  pendulum,  it  will  interfere  with  the  ac- 


Book  If^J  THE  PENDULUM.  279 


tion  of  the  compensation;  while  if  it  be  not,  the  effective  length 
of  the  pendulum  is  no  longer  determined  by  them.  Such  is  the 
plan  of  compensation  proposed  by  Dr.  Fordyce,  which  has  the 
advantage  of-  great  simplicity,  although  in  consequence  of  the 
defects  that  have  been  stated,  it  is  not  perfect. 

(2.)  The  rod  and  bulb  of  a  pendulum  being  separate  bodies, 
and  the  former  generally  passing  through  the  latter,  which  only 
rests  upon  a  projecting  part,  it  occurred  to  Graham,  that  were 
they  to  be  made  of  two  different  substances,  the  expansion  of  the 
rod  downwards,  might  be  compensated  by  the  expansion  of  the 
bulb  upwards.  There  is  not,  however,  a  sufficient  difference  in 
the  expansibilities  of  the  solid  metals,  to  allow  this  principle  to 
be  carried  into  effect  by  means  of  them.  But  mercury  expands 
about  16  times  as  much  as  steel;  and  hence,  could  the  rod  be 
made  of  the  latter  metal,  and  the  bulb  of  the  former,  in  a  ratio  of 
dimension  of  about  16:1,  a  compensation  might  be  effected. 
To  form  the  bulb,  the  mercury  is  placed  in  a  cylindrical  jar  of 
glass;  and  in  order  to  support  the  jar  from  beneath,  the  rod  is 
divided  into  two  branches,  connected  at  the  lower  end  by  a  plate, 
having  thus  a  shape  analogous  to  a  stirrup.  This  pendulum, 
called  the  Mercurial  Pendulum  of  Graham,  is  perhaps  the  most 
perfect  of  all  compensations.  Its  original  adjustments  are,  how- 
ever, more  difficult  than  those  of  some  others  we  shall  describe  ; 
and  if  it  should  be  removed  from  the  place  where  it  was  origi- 
nally adjusted,  the  experiments  must  be  again  repeated.  It  is, 
therefore,  only  used  infixed  observatories. 

To  investigate  the  relation  between  the  length  of  the  column  of 
mercury  in  the  vase  of  the  mercurial  pendulum,  and  the  whole 
length  of  the  rod  : 

Let  /  be  the  whole  length  of  the  steel  rod  ;  s  the  lineal  expan- 
sion of  steel  ;  y  the  unknown  length  of  the  column  of  mercury  ; 
m  the  cubic  expansion  of  mercury  ;  g  the  lineal  expansion  of 
glass.  Suppose  that  the  conditions  of  the  problem  require  the 
centre  of  gravity  of  the  mercury  to  remain  in  a  constant  position, 
which  would  be  true,  provided  the  position  of  the  centre  of  gravity 
of  the  rod  also  remained  constant.  The  distance  of  the  centre 
of  gravity  of  the  mercury  in  the  vase  from  the  bottom,  or  extre- 

y 

mity  of  the  pendulum,  is  ^,  and  the  distance,  L,  of  its  centre  of 
gravity  from  the  axis  of  suspension,  is, 

T-/     y. 
—  2  ' 

and  for  a  perfect  compensation,  this  expression  must  be  a  con- 
stant quantity,  however  /  and  g  vary  under  the  influence  of  tem- 
perature. 


880  APPLICATIONS  or  [Book  IV. 

If  r  be  the  radius  of  the  vase  that  contains  the  mercury  at  the 
standard  temperature  ;  V,  the  volume  of  mercury,  will  be 

V=*r2i/.  (308) 

At  t  degrees,  the  radius  of  the  vase  will  become 


y  will  also  vary  and  become  y'  ;  and,  at  the  same  time,  the  vol- 
ume of  mercury  will  become 


hence 

V(l+mO=irf*(l+  gtfT  ;  (309; 

dividing  this  by  the  equation  (308),  we  obtain 

(1+ffOY 
l+m<=        y         ;  (310; 

whence  we  have 


which  is  the  value  of  the  height  of  the  column  of  mercury  at  the 
new  temperature,  t.  But  as  g  is  but  a  small  quantity,  its  seconc 
power  may  be  neglected  without  any  sensible  error  ;  and  the  ex 
pression  will  become 

y'=y+y(m—2g)t.  (312 

The  constant  distance,  L,  will  at  the  same  temperature  be 

L=J_  f+Jd—  y;  (313 

and  substituting  the  value  oft/', 

;  (314 


and  as  this  must  be  equal  to  the  first  value,  the  variable  term  af 
fected  by  t  must  be  =0,  or 

Is—  |(w—  2o-)  =  0;  (315 

whence  we  have  for  the  value  of?/, 

y=l      2*      ;  (316 

m  —  2g  ' 

taking  the  exact  expansions  of  the  three  substances  between  th 
boiling  and  freezing  points,  we  have 

a=0.00124, 
w=0.01848, 
g=0.00088  ; 
and  substituting  these  values,  we  obtain 

I 
^-6/75' 


Book  IV.] 


THE  PENDULUM. 


281 


s  b  s  b 


1)    5   5s 


(3.)  Harison,  finding  that  Graham  had  failed  in  applying  the 
solid  metals  as  a  compensation  to  the  two  separate  parts  of  the 
pendulum,  abandoned  that  method,  and  sought  for  an  application 
of  them  to  the  rod  alone.      Inferring  from  experiments  on  the 
expansion  of  brass  and  steel,  that  their  relative  expansion  was 
nearly  in  the  ratio  of  3  :  5,  he  planned  a  frame  of  which  the  fol- 
lowing figure  will  give  an  idea.     It  was  composed  of  nine  paral- 
lel bars,  and  from  a  fancied  resemblance,  was  called  the  Gridiron 
Pendulum.       The   five   bars  marked  s,   are  of  steel,  the  four 
marked  6,  are  of  brass  ;  the  centre  rod  of  steel  is  fixed  at  top  to 
the  cross-bar,  connecting  the  two  contiguous  brass  rods,  but  slides 
freely  through  the  two  lower  bars  that  cross  its  direction.    This 
centre  rod  bears  the  bulb.     The  remaining 
rods  are  fastened  to  the  cross  pieces  at  both 
ends,  and  the  outer  and  upper  cross-piece 
is  attached  to  the  axis  of  suspension.     It 
will  be  apparent,  from  inspection,  that  the 
steel  rods  will,  in  their  expansion,  tend  to 
lengthen   the   pendulum,   while  the   brass 
rods  will,  in  theirs,  tend  to  shorten  it.     If 
these  two  expansions  exactly  counterba- 
lance each  other,  the  length  of  the  pendu- 
lum   will    remain   invariable.      The  four 
brass  rods  act  by  pairs,  and,  therefore,  as  if 
there  were   but  two  ;  and  no  more  than 
three  of  the  five  steel  rods  are  to  be  con- 
sidered, for  four  of  them  also  are  connected 
in  pairs.     Hence,  if  the  expansion  of  the 
sum  of  three  of  the  steel  rods  added  to  the 
expansion  of  the  rod  that  connects  the  grid- 
iron frame   to  the  axis  of  suspension  be 
equal  to  the  expansion  of  two  of  the  brass 
rods,  the  condition  of  compensation  is  ful- 
filled. 


To  investigate  the  lengths  that  should  be  given  to  the  brass 
rods  in  the  gridiron  pendulum  of  Harison  : 

Let  L  be  the  distance  from  the  axis  of  suspension  to  the  ex- 
treme end  of  the  pendulum  ;  and  let  ihe  condition  of  compensa- 
tion be  that  this  length  shall  remain  constant.  Let  /  be  the  com- 
mon length  of  the  outer  pair  of  steel  rods  ;  /'  that  of  the  inner  pair; 
a  the  distance  from  the  gridiron  frame  to  the  axis  of  suspension 
36 


APPLICATIONS  OF  [Book  IV. 

occupied  by  a  steel  rod  ;  and  b  the  length  of  the  steel  rod  that 
bears  the  bulb,  occupying  the  middle  of  the  frame.     Let  X  and 
/  be  the  several  lengths  of  the  two  pairs  of  brass  rods.  We  have 
from  the  construction  of  the  apparatus, 

L=o+  W+&—  X—  X'  ;  (317) 

if  we  call  the  expansions  of  brass  and  steel,  B  and  S,  we  have  for 
the  constant  value  of  L,  after  a  change  of  temperature  of/  de- 
grees, 

L=o+/+r+&—  x—  V 

+  [(a-f  W+&)  .  S—  (X+X')  .  B]  ;       (818) 
and  in  this  equation,  the  variable  term  affected  by  S  and  B  =0, 

(a+/-H'+6)S—  (X+X')B=0; 
but  we  have  from  our  first  equation, 


and  substituting  this  value  of  the  foregoing,  we  obtain 

(L+X+X)  S—  (X+X')  B=0  ;  (319) 

whence 

(X+X').  (B—  S)  = 

If  we  take  the  approximate  ratio  of  expansion  used  by  Hari- 
son,  we  have 

3L 


or  for  the  joint  length  of  the  two  pairs  of  brass  bars,  one  and  a 
half  times  the  whole  distance,  from  the  axis  of  suspension  to  the 
extremity  of  the  pendulum. 

These  investigations  may  serve  as  a  guide  in  the  construction 
of  these  two  species  of  pendulum,  but  they  are  obviously  inex- 
act. Nor  is  it  necessary  that  they  should  be  more  accurate  ;  for 
the  adjustment  of  each  must  finally  be  made  by  experiment  ;  and 
reference  must  be  had,  not  only  to  the  pendulum  itself,  but  to 
the  clock  ;  for  the  action  of  the  clock  on  the  pendulum  will  be 
affected  by  changes  of  temperature,  and  the  pendulum  must  meet 
this,  as  well  as  its  own  variations. 

In  the  gridiron  pendulum,  as  has  been  seen,  five  rods,  three 
of  steel  and  two  of  brass,  are  sufficient  for  the  purpose  of  com- 
pensation ;  the  other  four  are  added  for  the  purpose  of  making 
it  symmetric,  and  causing  the  line  of  direction  of  its  centre  of 
gravity  to  pass,  when  the  pendulum  is  at  rest,  through  the  centre 
of  magnitude  of  the  bulb. 

(4.)  The  expansion  of  an  alloy  of  S  pts  of  zinc  and  one  of  tin, 
is  to  that  of  steel  nearly  as  2  :  1.  Hence,  a  smaller  number  of 
bars  of  these  two  substances  will  furnish  a  compensation.  Thus 


Book  IV.\  THE  PENDULUM. 

the  pendulum  applied  by  Breguet  to  his  clocks,  is  composed  of 
no  more  than  five  rods,  three  of  steel  and  two  of  zinc;  two  of 
the  steel  rods  and  the  two  of  zinc  being  combined  in  pairs,  so 
that  it  may  be  considered  as  composed,  so  far  as  the  principle  of 
compensation  is  concerned,  of  no  more  than  three  rods. 

The  pendulum  of  Harison  has  been  improved  by  Troughton, 
who  has  substituted  for  the  two  pairs  of  brass  rods,  two  cylinders 
of  the  same  metal  sliding  one  within  the  other,  to  which  the  iron 
rods  are  attached.  The  principle  is  obviously  the  same,  but  it 
has  some  advantages  over  the  original  form,  inasmuch  as  it  is 
less  liable  to  external  injury,  and  the  brass  tubes  will  not  bend 
under  the  upward  pressure  of  their  expansion,  to  which  rods  of 
the  same  metal  are  liable. 

Such  are  the  more  important  of  the  forms  of  compensation 
pendulums.  The  number  and  variety  of  those  that  have  been 
proposed  is  very  great,  and  it  would  occupy  too  much  space  to 
enter  into  a  description  of  them.  Those  who  feel  a  curiosity  to 
examine  their  different  structures,  will  find  an  admirable  paper 
on  the  subject  by  Kater,  in  the  work  on  Mechanics,  which  bears 
the  joint  names  of  himself  and  Dr.  Lardner. 

283.  In  order  to  determine  the  intensity  of  gravity,  by  means 
of  the  pendulum,  it  becomes  necessary  to  measure  its  length; 
that  is  to  say,  to  determine  the  distance  between  its  axis  of  sus- 
pension and  centre  of  oscillation.      Two  principal  methods  are 
now  employed  for  this  purpose,  those  of  Borda  and  Kater. 

284.  In  Borda's  method,   the  experimental   pendulum,  from 
the  measure  of  which  the  length  of  the  second's  pendulum  is  to 
be  inferred,   is  composed  of  a  sphere  of  platinum,  suspended  by 
a  slender  wire  of  iron,  from  a  knife-edge  of  steel  resting  on  plane 
surfaces  of  polished  agate.     This  form,  employing  the  densest  of 
known  substances,   and  the  slenderest  wire  that  is  sufficient  to 
bear  it  with  safety,  approaches  as  nearly  as  possible  to  the  hypo- 
thetical simple  pendulum. 

Its  length,  considered  as  a  pendulum,  or  the  distance  between 
its  centres  of  suspension  and  oscillation,  is  determined  by  calcu- 
lation from  its  total  physical  length,  obtained  by  actual  measure- 
ment. To  effect  this  measurement,  the  pendulum  is  rendered 
stable,  by  screwing  up  from  beneath,  a  cup-shaped  vessel,  that 
just  catches  the  ball  of  the  pendulum,  without  bearing  any  part 
of  its  weight.  A  scale  of  iron  is  then  applied  to  it,  on  which  the 
physical  length  is  marked.  An  improved  method  consists  in 
screwing  up  from  beneath  a  plane  of  polished  steel,  until  it  just 
touches  the  platinum  sphere  ;  the  pendulum  is  then  removed, 
and  to  its  place  is  adapted  a  scale,  by  means  of  knife-edges  simi- 


284  APPLICATIONS  OP  [Book  IV. 

lar  to  those  of  the  pendulum.  This  scale  is  composed  of  two  parts, 
one  of  which  is  firmly  fastened  to  the  knife-edge,  and  is  shorter 
than  the  pendulum  ;  the  other  slides  upon  this,  and  is  moved  by 
a  screw.  The  scale  being  thus  placed,  the  moveable  part  is  de- 
pressed by  means  of  the  screw,  until  it  just  touches  the  steel  plate; 
the  length  of  the  two  portions  united,  that  is  to  say,  of  the  part 
fixed  to  the  knife-edge,  added  to  that  of  the  projection  of  the 
moveable  part,  is  of  course  just  equal  to  the  physical  length  of  the* 
experimental  pendulum. 

The  theoretic  length,  or  the  distance  between  the  axes  of  sus- 
pension and  oscillation,  is  next  deduced,  upon  the  principle  of  its 
being  a  sphere  suspended  by  a  line  void  of  weight,  by  the  for- 
mula (306), 


Such  at  least  would  be  the  principle,  were  the  wire  and  sphere 
the  only  parts  of  the  pendulum,  and  the  former  devoid  of  weight. 
As,  however,  neither  of  these  is  true,  particularly  as  parts  must 
be  adapted  to  attach  the  wire  to  the  knife-edge  and  to  the  sphere, 
a  much  more  complex  formula  must  be  used  in  practice.  We  re- 
fer for  this  to  the  "  Base  du  Systeme  Metrique,"  and  to  Delam- 
bre's  Treatise  on  Astronomy. 

In  the  original  apparatus  of  Borda,  the  length  of  the  experimen- 
tal pendulum  was  four  times  the  length  of  the  second's  pendulum. 
The  time  of  its  oscillation  was  determined  by  a  method  called 
that  of  Coincidences.  For  this  purpose,  the  pendulum  was  sus- 
pended upon  knife  edges,  resting  on  planes  of  agate,  in  front  of  a 
well-regulated  astronomical  clock,  having  a  compensation  pendu- 
lum. The  knife  edge  was  moved  upon  the  agate  planes,  until  the 
wire  of  the  experimental  pendulum,  as  viewed  through  a  small 
telescope,  placed  at  the  distance  of  12  or  15  feet  in  front  of  Iho 
clock,  exactly  coincided  with  the  centre  of  the  circle,  bounding 
the  lenticular  bulb  of  this  clock  pendulum.  This  point  was  mark- 
ed by  drawing  two  black  lirfcs  on  a  white  ground,  making  each 
an  angle  of  45°  with  the  horizon  ;  a  black  screen  was  so  placed 
as  to  hide  just  half  the  wire  of  the  experimental  pendulum.  The 
two  pendulums  being  set  in  motion,  an  observer  placed  at  the  tele- 
scope, would  see  the  wire,  and  the  point  marked  upon  the  clock 
pendulum  disappear  behind  the  edge  of  the  screen,  at  each  of 
their  alternate  oscillations.  If,  when  first  observed,  they  did  not 
pass  the  edge  of  the  screen  at  the  same  instant  of  time,  they  would, 
provided  the  one  were  not  exactly  four  times  as  long  as  the  other, 
gradually  approach,  until  both  would  disappear  at  the  same  in- 
stant. The  time  marked  by  the  clock  is  then  noted,  as  the  instant 
of  the  first  coincidence. 


Book  IV.]  THE    PENDULUM.  285 

It  is  usual  to  make  the  experimental  pendulum  a  little  longer 
than  four  times  that  of  the  clock ;  hence  the  former  makes 
a  little  less  than  one  oscillation  for  every  two  of  the  latter.  Af- 
ter the  interval  of  four  seconds,  the  wire  and  the  cross  will  be 
again  in  the  field  of  the  telescope  at  the  same  time,  but  the  cross 
will  precede  the  wire.  At  each  successive  interval  of  four  seconds, 
the  distance  at  which  they  pass  each  other,  will  increase  until  the 
interval  ofthe  times  of  their  respective  disappearances  amounts  to 
1 ".  After  this  they  will  approach,  until  they  again  pass  the  eye, 
and  disappear  behind  the  screen  at  the  same  instant,  which  is  no- 
ted as  that  of  a  second  coincidence.  During  this  interval,  the 
clock  pendulum  will  have  gained  two  oscillations  upon  the  experi- 
mental pendulum  ;  that  is  to  say,  the  number  of  the  oscillations  of 
the  experimental  pendulum,  will  have  been  one  less  than  half  the 
number  of  seconds  marked  by  the  clock ;  the  latter  number  is 
obtained  by  simple  inspection  ofthe  dial  ofthe  clock. 

The  observation  of  the  coincidences  is  continued,  until  the  ex- 
perimental pendulum  has  lost  too  much  of  its  mQtion  to  render 
them  easily  distinguishable,  and  the  record  of  the  times  is  col- 
lected in  a  set. 

2S5.  In  order  to  reduce  the  oscillations  performed  by  the  ex- 
perimental pendulum,  to  a  cycloid,  or  an  infinitely  small  circular 
arc  ;  the  extent  of  the  arcs  of  vibration  on  each  side  of  the  verti-  , 
cal,  are  observed  at  each  coincidence.  The  correction  is  applied 
upon  the  principle  of  the  formula,  (2S5),  in  whieli  the  last  terms 
of  the  series  of  (S3),  are  neglected.  This,  in  very  small  arcs, 
becomes  very  nearly 

sin.  2a 

-nr 

It  is  more  convenient,  however,  to  apply  the  correction  to  the 
whole  set,  in  which  case  the  mean  of  the  first  and  last  arcs  of 
vibration  may  be  taken.  But  as  the  arcs  decrease  in  fact  in  a  geo- 
metric progression,  a  more  correct  formula  has  been  calculated 
by  Borda,  which  is  as  follows  : 


j'\ 
ra —     v~  i  ~  j  ~*~-  v~     ^ ;        ^  '321} 

32  M  (log.  sin.  a — log.  sin.  a'  ' 
in  which  a  and  «'  are  the  greatest  and  least  amplitudes  ofthe 
oscillations;  and  M,   the  modulus  of  the  tables  of  logarithms, 
=2.30255509. 

The  pendulum  vibrating  in  air,  the  number  of  oscillations  it  is 
observed  to  make,  must  be  corrected  for  the  buoyajicy  of  the  air, 
which  is  done  by  the  formula  (307).  No  correction  has  hith- 
erto been  applied  for  the  variation  in  the  arc's  resistance,  growing 


286  APPLICATIONS  Off 

out  of  the  figure  of  the  pendulum,  detected  by  Bessel  and  Sabine, 
as  stated  in  §  279. 

As  the  temperature  of  the  apparatus  may  vary  during  the  series 
of  coincidences,  the  length  obtained  by  measurement  will  proba- 
bly not  be  the  same  as  that  at  which  anj'  one  of  the  coincidences 
has  occurred,  and  the  latter  will  differ  from  each  other.  The 
wire  being.  capable  of  expansion  and  contraction,  by  changes  of 
temperature,  it  will  be  necessary  to  reduce  the  whole  to  some 
standard  temperature.  So  also  must  the  length  of  the  rod,  by 
which  it  is  measured,  be  corrected  for  the  temperature  at  which 
that  part  of  the  operation  is  performed. 

The  corrections  for  the  expansion  by  temperature,  may  be  ob- 
tained from  the  tables  of  the  expansion  of  the  metals  that  are  to 
be  found  in  authors  on  physical  subjects  :  or,  they  may  be  ob- 
tained from  experiments  on  the  varying  rate  of  the  pendulum's 
own  oscillations,  at  different  degrees  of  heat.  The  last  method 
was  used  by  Sabine,  and  is  the  best,  inasmuch  as  the  actual  ex- 
pansion of  the  apparatus  is  obtained,  instead  of  the  mean  expansion 
of  the  species  of  substance  employed. 

When  the  length  of  the  experimental  pendulum  is  known,  and 
the  number  of  oscillations  it  performs  in  a  given  time,  say  in  a 
mean  solar  day,  is  determined  and  reduced  to  a  cycloidal  arc  and 
a  vacuum  ;  the  length  of  the  pendulum  that  would  vibrate  seconds, 
may  be  at  once  obtained  ;  for  it  has  been  shown,  §272,  that  the 
respective  lengths  of  pendulums,  are  inversely  as  the  squares  of 
their  numbers  of  vibration  in  equal  times;  hence,  as  there  are 
£6400  seconds  in  a  mean  solar  day,  if  L  be  the  lenyth  of  the  expe- 
r  mental  pendulum,  corrected  for  temperature  ;  N  Ihe  number  of 
vibrations  it  performs  in  a  mean  solar  day,  corrected  for  the  arc 
of  vibration,  and  for  the  buoyancy  of  the  air  :  the  length  /  of  the 
pendulum  that  vibratos  seconds,  will  be 


(86400)2  ' 

It  still  remains  that  the  length  thus  obtained  should  be  reduced 
to  the  mean  surface  of  the  earth,  or  to  the  level  of  the  sea.  The 
altitu/le  of  the  place  of  observation  above  this  surface,  must, 
therefore,  be  determined,  and  the  reduction  performed  by  the 
formula,  (293), 


The  form  and  nature  of  the  ground  will  influence  this  result  ;  and 
hence,  when  the  elevation  is  considerable,  the  length  thus  obtained 
will  not  be  that  which  would  be  found  at  the  level  of  the  sea.  It 
has  been  stated  by  Dr.  Young,  in  the  Transactions  of  the  Royal 


Book  IT.}  TUB    PENDULUM.  287 

Society  of  London,  that  if  the  station  be  upon  a  table  land  of  a 
density  equal  to  fds  of  the  mean  density  of  the  earth,  the  diminu- 
tion of  the  force  of  gravity  will  be  no  more  than  one-half  of  what 
is  due  to  the  height  above  the  level  of  the  sea;  and  that  in  the 
most  unequal  country,  there  will  be  at  least  £lh  to  be  deducted 
from  the  correction  obtained  by  the  above  formula,  (293).  In 
Sabine's  investigation,  this  correction  has  been  multiplied  by  the 
constant  coefficient,  0.6. 

It  has  also  been  discovered  by  Sabine,  whose  remark  has  been 
confirmed  by  Biot,  that  the  nature  of  the  ground  on  which  the 
experiment  is  performed,  affects  the  length  of  the  pendulum  in 
all  places.  The  attraction  of  gravitation  being  greater  upon  dense 
earthy  substances  than  it  is  upon  rare. 

As  an  instance  of  the  effect  of  elevated  ground,  we  shall  cite 
the  observations  of  Carlini,  at  Mount  Cenis.  The  pendulum 
measured  at  the  top  of  the  mountain,  and  reduced  to  the  level  of 
the  sea  by  the  usual  method,  had  a  length  in  French  measure  of 
0.^993708  ;  while  a  pendulum,  for  the  same  latitude  deduced  from 
the  pendulum  of  Bourdeaux,  would  not  have  been  longer  than 
0.^993498.  The  difference  of  0.^000210  is,  therefore,  due  to 
the  local  attraction.  This  observation  may  be  cited  as  being 
among  those  whence  the  density  of  the  earth  has  been  inferred. 

The  method  of  Borda  has  been  improved  by  Biot,  and  the  ap- 
paratus rendered  more  convenient.  The  length  of  the  experi- 
mental pendulum  has  been  reduced  to  one  that  differs  but  little 
from  that  of  the  pendulum  of  the  clock  :  a  copper  wire  has  been 
substituted  for  one  of  steel,  as  less  liable  to  rust;  and  the  whole 
apparatus  enclosed  in  a  glass  case,  to  render  it  less  exposed  to  the 
action  of  currents  of  air,  and  to  sudden  changes  of  temperature. 
With  this  diminished  length,  the  pendulum  that  moves  fastest 
still  gains  two  oscillations  upon  the  other,  between  each  coinci- 
dence. 

2S6.  The  method  of  Kater  is  founded  upon  the  principle, 
§276,  that  the  centres  of  suspension  and  pscillation  are  con- 
vertible points;  and  conversely,  that  if  a  pendulum  vibrate  in 
equal  times,  upon  two  different  parallel  axes,  one  of  these  has  the 
relation  to  the  other  of  the  axis  of  suspension  to  that  of  oscilla- 
tion. If  then  a  pendulum  be  taken,  into  which  two  knife  edges, 
turned  in  opposite  directions,  are  inserted  ;  if  the  distance  be- 
tween these  knife  edges  is  very  nearly  that  which  can  be  esti- 
mated to  exist  between  the  centres  of  oscillation  and  suspension  ; 
and  if  a  moveable  weight  be  adapted  to  the  rod,  this  weight  may 
be  so  adjusted  by  trials,  that  the  pendulum  shall  oscillate  in  equal 
times,  when  suspended  by  either  axis. 


288 


APPLICATIONS  OF 


[Book 


Cls 


The  form  of  the  pendulum  of  Kater  is  such  as  is  represented 
beneath. 

A  The  rod  AEFA',  is  a  bar  of  hammered  brass,  to 

which  is  adapted  the  bulb  B,  of  cast  brass,  and  of 
a  form  no  more  different  from  a  cylinder  than  is 
necessary  for  the  convenience  of  casting.  The 
knife  edges  made  of  wootz,  are  represented  at  a 
and  a' ;  they  are  formed  of  two  planes,  meeting  at 
an  angle  of  about  60°.  The  moveable  weight  is 
in  two  parts,  and  slides  on  the  bar  AA'.  The 
part  c,  is  capable  of  being  firmly  fixed  to  the  bar 
by  a  screw,  whose  head  is  represented  ;  the  part 
d,  is  attached  to  the  part  c,  by  a  screw  of  a.  fine 
thread,  by  means  of  which  a  slow  motion  may  be 
given  to  d  after  c  is  made  fast.  The  pendulum  is 
brought  nearly  to  its  adjustment,  by  sliding  the 
whole  weight  along  the  rod  ;  c  is  then  firmly  fast- 
ened by  its  screw,  and  the  adjustment  is  completed 
by  the  slow  motion  of  d. 

The  portions  E  and  F,  of  the  rod  A  A',  were 
in  the  original  apparatus  of  Kater,  made  of  black- 
ened wood. 

The  observations  of  the  coincidences,  by  means 
of  which  the  pendulum  is  adjusted,  and  whence 
the  value  of  its  oscillations  is  determined  for  cal- 
culation, when  adjusted,  are  made  as  follows  : 
The  clock  pendulum  being  at  rest,  a  small  tele- 
scope is  placed  directly  in  front  of  it,  and  the  ex- 
perimental pendulum  is  suspended  in  such  a  man- 
ner that  one  of  its  blackened  terminations,  just 
hides  a  white  circular  spot,  drawn  upon  the  lens 
of  the  clock  pendulum,  and  concentric  with  it. 
The  diaphragm  of  the  telescope  has  two  plates 
with  vertical  edges ;  these  are  pressed  forward  by 
screws,  until  they  appear  in  optical  contact  with 
the  blackened  bar.  When  the  pendulums  are  set 
in  motion,  during  the  greater  part  of  the  oscilla- 
^^^^^  tions,  the  bar  and  the  white  spot  will  both  be  seen  ; 

but,  from  time  to  time,  for  a  few  contiguous  oscil- 
lations, the  bar  will  wholly  hide  the  white  spot.  Kater  took, 
as  the  instant  of  coincidence,  the  beat  of  the  pendulum  that  fol- 
lowed the  first  passage  in  which  no  part  of  the  white  spot  was 
seen.  This  method  is  objectionable,  inasmuch  as  the  same  ob- 
server will,  under  different  circumstances  of  light,  continue  to 


Book  IV ,]  THE    PENDULUM.  289 

see  the  spot  a  longer  or  shorter  time  before  it  is  wholly  ob- 
scured. The  tact  of  observation,  even  in  the  same  observer,  will 
differ  at  different  times,  and  will  improve  by  practice;  while, 
when  the  experiments  of  different  observers  are  compared,  a 
marked  difference  will  be  found  to  exist  in  their  powers  of  vi- 
sion. 

For  these  reasons,  Sabine,  in  his  experiments,  adopted  the  plan 
of  observing,  not  only  the  disappearance,  but  also  the  reappear- 
ance of  the  white  disk  ;  noting  the  second  succeeding  the  former, 
and  preceding  the  latter  :  the  mean  of  the  two  was  taken  as  the 
instant  of  coincidence.  The  author  had  a  good  opportunity  of 
testing  the  value  of  these  two  different  methods,  when  associated 
with  Sabine,  in  the  experiments  to  determine  the  length  of  the 
pendulum  at  New- York.  Had  the  experiments  made  by  the  two 
observers  been  compared  upon  the  plan  of  Kater,  a  considerable 
discrepancy  would  have  ensued  ;  but  when  compared  by  the  me- 
thod of  Sabine,  the  results  were  nearly  identical. 

The  length  of  the  experimental  pendulum,  in  Kater's  method, 
is  determined  directly,  without  reference  to  calculation,,  by  mea- 
suring the  distance  between  the. knife  edges.  This  is  effected 
by  means  of  a  scale  furnished  with  powerful  microscopes  ;  to  one 
of  these  a  micrometer  is  adapted.  With  this  apparatus,  the 
10,000th  part  of  an  inch  becomes  a  measurable1  quantity.  The 
method  of  Kater  requires  the  same  corrections  and  reductions  as 
that  of  Borda;  thus  the  time  of  oscillation  must  be  corrected  for 
the  arc  of  vibration,  and  for  the  buoyancy  of  the  air  ;  the  length 
must  be  corrected  for  the  temperature,  and  the  second's  pendu- 
lum, calculated  from  the  observations,  reduced  to  the  level  of  the 
sea. 

The  method  of  Kater  is  liable  to  one  objection,  namely,  that 
in  conformity  with  the  views  of  Bessel,  it  will  be  unequally  re- 
sisted by  the  air,  when  suspended  by  its  different  knife-edges. 
A  modification  of  the  apparatus  proposed  by  Bailey,  is  less  liable 
to  this  objection  :  this  pendulum  is  a  simple  bar,  without  a  bulb, 
and  the  adjustment  is  effected  by  filing  away  those  portions 
whose  weight  is  in  excess. 

287.  In  order  to  determine  the  measure  of  the  force  of  gravity 
at  any  given  place,  when  the  length  of  the  second's  pendulum  is 
known,  we  have  the  formula  (289) 

g=l«*. 

At  New-York,  the  length  of  the  pendulum,  or  /,  is  39.1  in, 
nearly,  or  3  ft.  2583,  hence 

g=32  ft.  1576, 
or  nearly  32|  feet. 
37 


290  APPLICATIONS  OF  [Book  IV. 

A  heavy  body  will,  in  consequence,  fall  in  vacuo  during  the 
first  second  of  time,  §  271,  through  a  space  equal  to  16^  feet. 

288.  It  has  been  stated  that  the  apparent  intensity  of  gravity, 
or  the  difference  between  its  absolute  force,  and  the  diminution 
growing  out  of  the  earth's  rotation,  may  be  immediately  deduced 
from  a  measure  of  the  second's  pendulum.     In  the  method  of 
Borda,  the  experimental  pendulum  is  made  to  vibrate  in  the  se- 
veral places  in  which  it  is  desired  to  ascertain  this  quantity  ;  but 
as  the  length  of  the  suspending  wire  may  vary,  it  becomes  ne- 
cessary to  go  through  the  whole  process  at  each  station. 

In  the  method  of  Kater,  as  the  distance  between  the  knife 
edges  is  invariable,  except  by  changes  of  temperature,  the  influ- 
ence of  which  is  known,  one  careful  measurement  will  suffice  for 
any  number  of  stations.  /The  original  pendulum  may,  therefore, 
be  carried  from  station  to  station,  and  its  coincidences  observed. 
A  direct  comparison  between  those  observed  at  different  places, 
givesan  immediate  determination  of  the  length  of  the  pendulum 
that  would  oscillate  in  a  second,  at  the  several  different  stations. 

The  method  has  been  still  farther  improved  by  its  author.  A 
pendulum,  having  but  one  knife  edge  at  the  usual  point  of  sus- 
pension, is  suspended  in  front  of  the  clock,  in  the  place  where 
the  original  experiment  was  made,  with  the  pendulum  with  con- 
vertible axes.  The  rate  of  its  oscillations  being  determined,  its 
length  can  be  calculated  by  the  formula  (290),  from  that  of  the 
original  pendulum.  It  may  then  be  carried  to  other  places,  and 
the  length  of  the  pendulum  of  the  place  determined  from  the  rate 
of  its  oscillations,  in  the  same  manner.  In  this  way,  Kater  him- 
self determined  the  length  of  the  second's  pendulum  at  the  more 
important  stations  of  the  British  Trigonometrical  Survey.  Sabine 
has  since  employed  the  same  method,  at  a  variety  of  stations, 
from  Ascension,  in  lat.  7°,  55',  48",  S.  ;  to  Spitsbergen,  in  lat. 
79°.  49'.  58".  N. 

This  part  of  Kater's  method,  as  applicable  to  observations,  at 
different  places,  is  much  more  convenient  than  that  of  Borda, 
even  as  improved  by  Biot.  It  may  also  be  performed  in  a  much 
less  time  ;  thus,  for  instance,  Biot  was  engaged  for  three  months 
at  Unst,  in  completing  his  measure  of  the  pendulum,  while  Kater 
effected  his  in  three  weeks. 

289.  The  relation  between  the  lengths  of  the  pendulum,  at 
different  places,  may  also  be  determined  by  means  of  a  clock  fur- 
nished with  a  pendulum  whose  rod  is  not  liable  to  have  its  di- 
mensions changed  by  transportation,  except  in  consequence  of 
variations  of  temperature.     Such  a  clock  was  used  in  the  first 


Book 


THE  PENDULUM. 


291 


expeditions  of  Parry  and  Ross,  and  the  absolute  length  was  ascer- 
tained, by  comparison  with  the  original  experiment  of  Kater. 

The  method  of  Kater  is  still  imperfect,  inasmuch  as  the  length 
determined  at  the  original  station,  and  therefore  at  all  others, 
still  rests  upon  his  own  single  experiment  ;  and  it  has  not  yet 
been  ascertained,  how  far  it  is  possible  for  the  same  experimenter, 
or  another  equally  well  qualified,  to  reproduce  an  identical  re- 
sult. Until  this  question  be  settled,  i^  must  remain  questionable 
whether  the  differences  in  the  measure  of  the  pendulum  at  the 
same  place,  by  the  two  different  methods  of  Borda  and  Kater, 
arise  from  the  methods  themselves,  or  are  involved  in  the  origi- 
nal determination  on  which  the  results  of  the  latter  method  are 
founded. 

290.  By  the  use  of  these  two  metho  Is,  the  pendulum  has  been 
measured  in  various  places  in  both  hemispheres,  by  Kater,  Sa- 
bine,  Biot,  Bouvard,  Matthieu,  Arago,  Chaix,  Freycinet,  and 
Duperrey.  Some  of  these  results  are  to  be  found  in  the  following 

TABLE. 


STATION. 

NORTH 
LATITUDE. 

ill 

£.00 

3  e  :£ 

PENDULUM. 

At  fhe 

Station. 

Reduced  to 
the  level  of 
the  Sea. 

St.  Thomas,    .     .     . 
Maranham,      ... 
Sierra  Leone, 
Trinidad,    .... 
Jamaica,     .... 
Formontera,     .     .     . 
New-  York, 
Bourdeaux,      .     .     . 
Paris            .... 

0".24'.41" 
2.31  .43 
8  .29  .28 
10.38.56 
17.56.07 
38.39.56 
40  .42  .43 
44  .50  .26 
48.50  .14 
51  .31  .08 
55  .58  .39 
60.45.26 
70  .04  .05 
79  .49  .58 

21ft 
77 
190 
21 
9 
606 
67 
56 
230 
921 
69 
30 
29 
21 

39.02069 
39.01197 
39.01954 
39.0187^ 
39.03508 
39.09176 
39.10153 
39.11282 
39.12843 
39.13908 
39.15540 
39.17145 
39.19512 
39.21464 

39.02074 
39.01214 
39.01997 
39.01884 
39.03510 
39.09325 
39.10168 
39.11295 
39.12894 
39.13929 
39.15546 
39.17151 
39.19519 
39.21469 

London,      .... 
Leith,    

TTust 

Harnmerfest,  .     .     . 
Spitsbergen,     .     .     . 

292 


APPLICATIONS  OP 


A  part  of  the  observations  that  have  been  made  in  the  south- 
ern hemisphere  will  be  found  in  the  following 

TABLE. 


STATION. 

SOUTH 
LATITUDE. 

PENDULUM  REDUCED 
TO  THE  LEVEL  OF 
THE  SEA. 

Ascension, 

7l>.55'.48" 

39.02410 

Bahia,         .... 
Isle  of  France,      .     . 
Port  Jackson,       .     . 
Malouine  Islands 

10.38.56 
20.09.19 
33.51.39 
51  .31  .44 

39.02435 
39.04793 
39.08049 
39.13695 

The  general  inference  of  Sabine,  from  a  combination  of  his 
own  experiments  with  those  of  Kater^  and  those  made  at  the 
stations  of  the  French  trigonometric  survey,  is  that  the  length  of 
the  equatorial  pendulum  =39  in.  01569  ;  the  increase  of  its  length 
from  the  equator  to  the  pole  =O  in.  20227.  The  formula,  (296), 
for  the  length  /',  at  any  intermediate  latitude  L,  becomes  (296), 

/'=39.01569-|-0.20227  sin.2L. 

A  more  recent  French  deduction,  into  which  the  observations 
of  Duperrey  and  Freycinet  enter,  gives 

J'=39.01741  +.23505  sin.=  L. 

The  result  obtained  by  Sabine  gives,  for  the  oblateness  of  the 
terrestrial  spheroid,  oj^;  and  for  the  centrifugal  force  at  the 
equator,  T-J-Tth  part  of  the  whole  force  of  gravity.  The  last  de- 
duction gives  for  the  centrifugal  force,  TJ7.  The  centrifugal 
force  is  usually  stated  at  T4F,  which  lies  between  the  above  in- 
ferences ;  and  this  is  the  value  that  we  shall  employ. 

291.  It  had,  until  the  discovery  of  the  influence  of  local  at- 
traction by  Sabine,  been  generally  concluded,  that  the  pendulum 
vibrating  seconds  in  a  given  latitude,  at  the  level  of  the  sea,  was 
a  constant  and  invariable  quantity.  Its  length  is  also  capable  of 
comparatively  easy  determination  ;  as  all  the  observations  con- 
nected with  its  measure  may  be  made  within  the  space  of  a  few 
weeks.  Hence  it  has  been  proposed  as  a  standard  of  measure. 
Were  the  first  inference  true,  and  were  the  reduction  to  the  level 
of  the  sea  independent  of  local  influence,  no  better  method  could 
well  be  devised  for  the  purpose  of  re-establishing  standards  of 
lineal  measure  that  have  been  lost,  or  are  suspected  of  being  al- 
tered by  age.  It  has  also  been  proposed  to  use  the  length  of  the 
pendulum  not  only  as  the  standard,  but  as  the  unit  of  lineal  mea- 
sure. This,  however,  is  objectionable,  except  in  the  case  where 


:i    '  -, 

Book  IV.]  THE  PENDULUM.  293 

the  customary  unit  of  a  country  differs  but  little  from  the  length 
o;"  the  pendulum.  Such  is  the  case  with  the  measure  of  Denmark  ; 
aid  hence,  underthe  auspices  of  Schumacher,  a  system  of  weights 
aad  measures  has  been  formed  in  that  country,  of  which  the  pen- 
dilum  forms  both  the  unit  and  the  standard. 

In  cases  where  the  difference  between  the  unit  in  actual  use 
and  the  length  of  the  pendulum  is  considerable,  it  is  better  to 
retain  the  ancient  unit,  and  define  its  relation  to  the  pendulum. 
For  this  purpose,  it  will  be  evident  from  what  has  been  said  in 
relation  to  the  influence  of  local  circumstances,  that  it  will  not 
be  safe  to  use  the  pendulum  of  a  given  latitude;  but  that  the  only 
admissible  method  is  to  take  the  pendulum  measured  in  a  par- 
t  cular  place  as  the  standard. 

When  a  unit  of  lineal  measure  has  been  defined,  in  relation 
t»  some  standard  existing  in  nature,  its  square  will  serve  as  a 
uiit  of  measures  of  surface ;  its  cube,  or  some  aliquot  part, 
as  the  unit  of  measures  of  capacity  ;  and  the  weight  of  its  cube, 
filed  with  pure  water  at  some  given  temperature,  will  furnish  a 
Uiit  of  measures  of  weight.  It  has,  however,  been  found  more 
eisy  to  determine  the  weight  of  water  that  a  measure  of  capacity 
wll  hold,  than  to  ascertain  its  cubic  contents;  and  hence,  in 
sone  systems,  the  unit  of  capacity  has  been  defined  by  declaring 
what  number  of  the  units  of  weight  it  shall  contain. 

292. -This  being  premised,  we  shall  proceed  to  describe  the 
principles  upon  which  a  reform 'has  been  effected  in  the  standards 
of  England,  and  of  the  State  of  New- York,  in  both  of  which  the 
pendulum  has  been  assumed  as  the  basis. 

The  standard  of  measure  in  Great  Britain  is  the  pendulum,  vi- 
brating seconds  in  a  cycloidal  arc,  in  a  vacuum,  and  at  the  level 
of  the  sea,  in  the  latitude  of  London,  51°  31'  OS"  N. 

The  unit  of  measures  of  length  is  the  Yard  of  such  magnitude 
that  the  pendulum  shall  measure,  39  in.  13929.  The  yard  is  di- 
vided into  three  feet ;  the  foot  into  twelve  inches  ;  and  for  cloth 
measure  the  binary  subdivision  is  permitted.  Greater  measures 
of  length  are  multiples  of  the  yard,  derived  as  in  the  ancient 
system. 

The  square  of  the  yard,  or  of  any  other  unit  of  length,  may  be 
used  as  a  unit  of  superficial  measure. 

The  standard  temperature,  to  which  measures  of  length  or  sur- 
face are  to  be  reduced,  is  62°  of  Fahrenheit's  thermometer,  and 
the  material  of  which  the  standard  yard  is  made  is  brass. 

The  unit  of  weight  is  the  Troy  Pound,  of  such  magnitude  that 
a  cubic  inch  of  water,  at  62°,  weighs  252  grs.  458,  there  being 
5760  grs.  in  this  pound. 


294  APPLICATIONS  OF  [Book  1 1 '•  . 


\ 


The  avoirdupois  pound  is  also  used,  and  isdefinedas  beingequpl 
to  7000  grs..Troy. 

The  unit  of  measures  of  capacity  is  the  Gallon,  which  is  a  vefr- 
sel  thai  holds  exactly  ten  avoirdupois  pounds  of  water,  at  tBe 
temperature  of  62°. 

The  bushel  holds  eighty  pounds  of  water  at  the  same  tempe- 
rature. 

293.  The  standard  of  the  state  of  New-York  is  the  pendulum 
vibrating  seconds  in  a  cycloidal  arc,  and  in  a  vacuum  in  Colum- 
bia College  in  the  city  of  New-York. 

The  unit  of  lineal  measure  is  the  Yard,  which  is  of  such  mag- 
nitude as  to  bear  to  the  pendulum  the  proportion  of  1,000000  to 
1,086158. 

Its  usual  subdivisions  are  allowed  to  be  employed  ;  and  is 
standard  temperature  is  that  of  melting  ice.  It  is  identical  will 
the  present  British  standard  yard,  which  has  been  restored  in  is 
magnitude  to  that  used  previous  to  the  revolution,  and  whi<p 
had  continued  in  use  in  the  State  of  New-York  ;  but  in  the  coir 
parison,  each  is  to  be  taken  at  its  own  standard  temperatur . 
The  standard  temperature  of  the  English  system  is  62°  Fahre  - 
heit ;  of  that  of  the  State  of  New-York,  32°. 

The  unit  of  measures  of  weight,  is  the  Avoirdupois  Poutd, 
of  such  magnitude  that  a  cubic  foot  of  pure  water,  at  its  maxi- 
mum density,  shall  weigh  1000  oz.  or  G2£lbs. 

The  unit  of  dry  measures  of  capacity,  is  the  Gallon,  a  vessel 
of  such  magnitude  as  to  hold  exactly  lOlbs.  of  pure  water,  at  its 
maximum  density.  The  bushel,  therefore,  holds  80lbs. 

The  unit  of  liquid  measure  is  also  a  gallon,  containing  eight 
pounds  of  distilled  water,  at  its  maximum  of  density.  The 
adoption  of  this  unit,  was  a  deviation  from  the  original  plan, 
which  contemplated  but  one  set  of  measures  of  capacity  for  so- 
lids and  fluids,  and  it  has  impaired  the  symmetry  of  the  system. 

294.  To  the  English  system,  it  is  to  be  objected  :   that  it  as- 
sumes for  its  standard  the  pendulum  of  a  particular  latitude,  which 
will  not  be  constant,  in  consequence  of  local  circumstances  ;  that 
the  determination  on  which  the  length  of  this  standard  has  been 
performed  in  a  private  building,  the  house  of  Mr.  Brown  ;  that 
it  retains  two  units  of  weight,  of  the  same  denomination,  but  of 
different  magnitudes;  and  that  its  standard  temperature  is  wholly 
arbitrary,  founded  on   no  natural  phenomenon,  and   dependent 
upon  a  conventional  thermometric  scale.     The  mode  of  defining 
its  unit  of  weight,  moreover,  involves  a  fractional  quantity,  ami 
the  bulk  of  water  employed  in  the  determination,  namely,  a  cu- 
bic inch,  is  too  small. 


Book  IT.]  THE  PENDULUM.  295 

To  the  system  of  the  State  of  New- York  none  of  these  objec- 
tions apply,  except  so  far  as  relates  to  the  double  system  of  mea- 
sures of  capacity.  The  standard  is  the  pendulum  of  a  particular 
place;  and  that,  so  far  as  is  known,  is  invariable;  that  place  is 
a  public  building,  readily  accessible  ;  the  standard  temperature* 
are  defined  by  physical  states  of  water,  in  respect  to  which  there 
can  be  no  error,  and  which  are  independent  of  thermometric 
scales.  The  unit  of  weight  is  determinable  from  a  bulk  of  water 
of  sufficient  magnitude. 

295.  The  French  system  of  weights  and  measures  has  for  its 
standard  a  quadrant  of  the  meridian.  The  unit  of  measures  of 
length  is  the  Metre,  which  is  a  ten  millionth  part  of  the* quad- 
rant. 

The  unit  of  superficial  measure,  is  the  Are,  a  surface  10  metres 
each  way,  or  100  square  metres. 

The  unit  of  measures  of  capacity,  is  the  Litre,  a  vessel  con- 
taining the  cube  of  a  tenth  part  of  the  metre. 

The  unit  of  weight  is  the  Gramme  which  is  equal  to  the  weight 
of  the  cube  of  the  hundredth  part  of  the  metre,  filled  with  distil- 
led water,  at  its  maximum  of  density,  or  to  the  1000th  part  of 
the  weight  of  a  litre  of  Water. 

The  standard  temperature  of  the  measures  of  length  is  that  of 
melting  ice. 

The  whole  of  the  divisions  and  multiples  of  the  units  were  de- 
cimal, and  the  principal  of  nomenclature  adopted,  was  to  prefix 
the  Greek  numerals  to  the  decimal  multiples,  and  the  Romaa 
numerals  to  the  decimal  subdivisions  of  the  units. 

Thus  the  measures  of  length  are, 

Myriametre  =  10000  metres. 
Kilometre  =  1000  metres. 
.  Hectometre  =  100  metres. 
Decametre  =  10  metres. 
Metre  =  1  metre. 

Decimetre     =          *-$  metre. 
Centimetre   =       ,1^  metre. 
Millimetre    =      ,  ^  metre. 
The  measures  of  Surface  are 

Hectare  =  10.000  sq.  metres. 

Are          =        100  sq.  metres. 

Centiare  =  1  sq.  metre. 

The  measures  of  Capacity  are 

Kilolitre      =  1000  litres. 

Hectolitre  —     100  litres. 

Decalitre    =       10  litres. 

Litre  =        1  litre. 


296  APPLICATIONS  OF  [Book  IV. 

Decilitre     =      TV  litre. 
Centilitre    =     r£¥  litre. 
The  weights  are 

Myriogramme  =  10000  grammes. 

Kilogramme     =  1000  grammes. 

Hectogramme  =  100  grammes. 

Decagramme  =  10  grammes. 

Gramme          =  1  gramme. 

Decigramme    =  TV  gramme. 

Centigramme  —  T^  gramme. 

Milligramme    =  ,  7V  F  gramme. 

296.  No  system  can  be  imagined  more  perfect  and  beautiful,  in 
a  scientific  point  of  view,  than  this  system  of  the  French  nation. 
It  is  founded  upon  a  standard  existing  in  nature,  and  invariable, 
and  which  is  susceptible  of  determination  to  such  a  degree  of  ex- 
actitude, that  no  probable  error  that  can,  in  the  present  state  of 
science,  be  committed  in  the  measure  of  degrees,  will  affect  the 
small  fraction  of  the  standard  that  forms  the  unit  of  length. 
From  its  decimal  division,  it  is  exactly  consistent  with  our  usual 
system  of  arithmetic;  and  its  nomenclature  is  systematic,  and  of 
easy  recollection.  Still  it  is  not  without/ault,  even  in  a  scientific 
point  of  view.  The  measures  of  length  are  incapable,  for  instance, 
of  application  to  astronomic  purposes,  in  which  we  use  the  semi- 
diameter  of  the  earth,  and  not  its  quadrant  as  the  unit ;  and  these 
two  magnitudes  are  incommensurable.  Neither  are  we  aware 
that  a  measure  of  the  meridian  in  other  countries,  particularly 
in  our  own  hemisphere,  would  reproduce  the  same  magnitude 
for  the  quadrant  that  was  obtained  in  France.  The  measure  of 
a  sufficient  arc,  whence  to  determine  the  length  of  a  quadrant,  is 
an  operation  of  grest  expense,  and  would  occupy  a  long  time. 
Hence,  in  presenting  the  types  of  the  units  to  the  National  Assem- 
bly, the  commission  propose  to  verify  them,  if  suspected  of  alter- 
ation, and  reproduce  them,  if  lost,  by  reference  to  the  pendulum 
of  the  Observatory  of  Paris;  thus  recurring  to  the  same  natural 
standard  that  had  been  rejected  by  them  in  the  outset.  The 
metre  is,  therefore,  after  all  the  labour  that  was  expended  in  its 
determination,  no  more  than  a  conventional  length,  whose  rela- 
tion to  the  second's  pendulum  of  a  particular  place  is  well  deter- 
mined. It  has  also  been  found  impracticable  to  introduce  the 
decimal  division  into  the  measure  of  angles;  and  after  strenuous 
attempts  for  that  purpose,  and  the  laborious  construction  of  new 
tables,  even  the  astronomers  of  France  have  returned  to  the  an- 
cient division  of  the  circle. 

The  objections,  in  a  practical  point  of  view,  are  more  nume- 
rous, and  have  been  found  insuperable.    Thus,  however  well-cal- 


Book  IV.~\  THE  PENDULUM.  297 

culated  for  scientific  purposes,  and  even  for  those  of  commerce, 
the  decimal  multiples  of  the  unit  may  be,  decimal  subdivisions 
have  been  found  unsuited  for  the  purpose  of  retail  traffic  ;  for  this 
object  no  other  than  a  binary  system  can,  with  convenience,  be 
used.  In  fact,'  in  the  subdivisions  of  the  unit,  no  other  method 
appears  to  be  consistent  with  nature  ;  and  those  systems  which  are 
founded  on  division  by  two,  appear  to  defy  any  attempts  to  alter 
them.  Thus  the  system  of  money  in  the  United  States,  which  is 
strictly  decimal,  is  only  used  in  written  calculations;  while  the 
old  binary  division  of  the  Spanish  dollar  is  retained  in  all  retail 
operations,  in  spite  of  the  barbarous  nomenclature  that  is  applied 
to  it  in  some  of  the  States.  Several  of  the  units  of  the  French 
system,  or  their  decimal  divisions  and  multiples,  are  unsuited  to 
ordinary  transactions  ;  subdivisions  suitable  to  these  were,  there- 
fore, first  introduced  clandestinely,  and  afterwards  sanctioned  by 
law.  Thus  a  measure  of  the  length  of  about  a  foot,  is  the  most 
convenient  for  many  mechanical  uses  ;  and  for  this  purpose  a 
measure  of  the  third  part  of  a  metre  was  formed,  called  the  Me- 
trical Foot,  to  which  the  ancient  duodecimal  subdivision  was  ap- 
plied. The  kilogramme  differing  but  little  from  two  ancient 
pounds,  its  half  has  become  the  unit  of  weight  in  actual  use,  and 
is  called  the  Metrical  Pound  ;  to  this,  also,  a  binary  division  has 
been  applied,  and  the  decimal  system  in  the  desending  scale  has 
not  only  failed  in  being  introduced  into  commerce,  but  has  been 
abolished  by  authority. 

Thus  there  are  at  present  in  France,  in  fact,  three  diverse  sys- 
tems ;  the  ancient,  which  is  not  wholly  abandoned  ;  the  decimal 
system  of  the  commission  ;  and  a  system  derived  from  the  latter, 
to  which  the  ancient  names,  and  many  of  the  ancient  subdivisions 
are  applied. 

The  system  proposed,  and  partially  introduced  by  the  French 
philosophers,  may,  therefore,  be  considered  as  a  splendid  failure, 
worthy  however  of  a  better  fate,  from  the  scientific  skill  with 
which  the  operations  connected  with  it  were  executed.  It  is  also 
memorable  for  the  light  it  has  thrown  on  all  analogous  processes, 
and  the  actual  benefit  the  researches,  in  respect  to  it,  have  confer- 
rld  upon  physical  science.  Warned  by  the  example  of  the  French, 
the  British,  Danish,  and  American  governments,  have  wisely  re- 
stricted themselves  to  the  verification  of  the  measures  in  actual 
use,  and  their  restoration  to  their  true  dimensions.  The  two  for- 
mer, and  the  state  government  of  New-York,  have  referred  them 
to  the  pendulum,  a  standard  existing  in  nature,  determinate,  and 
easily  determinable. 

The  determination  of  units  of  measures  of  capacity,  and  of 
weight,  from  standards  existing  in  nature,  involves  certain  nice- 

38 


293  APPLICATIONS  OF  [Book  IV. 

ties  founded  on  the  mechanics  of  fluid  bodies.  These  will  be  fully 
explained  in  a  subsequent  part  of  this  work. 

297.  Among  the  applications  of  the  theory  of  thetpendulum, 
may  also  be  classed  the  principle  of  the  calculations  by  means  of 
which  the  density  of  the  earth  is  ascertained  from  the  experiment 
of  Cavendish,  §  91.  The  balls  attached  to  the  balance  being  set 
in  motion,  and  caused  to  oscillate  by  the  attraction  of  the  masses 
of  lead  presented  to  them,  may  evidently  be  considered  as  a  hori- 
zontal pendulum  actuated  by  that  attractive  force.  By  comparing 
the  length  of  this  pendulum  with  that  of  a  common  pendulum, 
that  would  oscillate  in  the  same  time,  we  may  obtain  the  relation, 
between  the  attractive  force  of  the  spheres  of  lead,  and  of  the  earth, 
considered  as  a  sphere.  Trie  equation  that  expresses  this  relation 
may  be  thus  investigated  : 

Let  us  consider  the  bodies  attached  to  the  extremities  of  the 
horizontal  bar  of  Cavendish's  apparatus,  as  if  their  masses  were 
collected  in  a  single  point,  and  abstract  the  mass  of  the  bar  itself, 
so  that  each  arm  of  the  balance  may  be  considered  as  a  simple 
pendulum. 

Let  the  length  of  the  arm  =a;  the  distance  of  the  centre  of 
gravity  of  the  attracting  mass,  from  the  point  of  suspension  of  the 
Dar?  =c  ;  the  angular  distance  between  the  end  of  the  bar,  and 
the  centre  of  gravity  of  the  attracting  mass,  at  the  time  motion 
begins,  =a  ;  their  mean  angular  distance  in  the  oscillations  =/3  ; 
the  measure  of  the  attractive  force  exerted  at  the  distance  of  the 
unit  of  lineal  measure  in  which  the  distances  are  estimated,  by  a 
mass  whose  magnitude  is  equal  to  the  unit  of  weight  in  which  the 
masses  are  estimated,  =/;  the  distance  between  the  attracting 
mass,  and  the  end  of  the  bar  at  the  time  motion  begins,  =c. 

Call  the  measure  of  the  attractive  force  of  the  earth,  g-,  the  ra- 
dius of  the  earth,  R,  and  its  mass  m,  we  shall  have  for  the  value 
of  g,  in  t«rms  of  m,/,  and  R, 

•'',,.;  <"&•  M 

We  should  in  like  manner  have  for  the  value  of  the  attractive  force 
of  the  mass  employed  in  the  experiment,  provided  its  distance 
from  the  end  of  the  pendulum  were  constant,  in  terras  of  its  mass 
wi',  of  the  force/,  and  the  distance  c, 


But  this  force  does  not  act  directly  :  it  must,  therefore,  be  decom- 
posed into  two,  one  of  which  is  perpendicular,  the  other  parallel 
to  the  bar  that  oscillates.  The  former  alone  acts,  and  its  value 
will  be  from  §  13, 

w»'/o  sin.  a 


Book  77^.]  THE    PENDULUM.  299 

We  are  next  to  consider  that  this  is  not  a  constant  force,  but  that 
it  acts  with  the  least  intensity  at  the  time  motion  begins,  and  in- 
creases until  the  bar  approaches  nearest  to  the  attracting  mass. 
At  this  point,  the  torsion  of  the  wire  of  suspension  overcomes  the 
motion,  and  causes  the  bar  to  return.  When  the  deflection  =/3, 
the  two  forces  balance  each  other,  and  at  this  time  we  have  the 
mean  value  of  the  attractive  force  which  will  therefore  be  a  func- 
tion of  f3.  And  we  may  without  ^investigation  assume,  what  might 
be  shown  by  a  rigorous  analysis,  that  it  is  inversely  proportioned 
to  (3,  or  that  to  find  the  value  of  the  attractive  force  that  acts  on 
the  balance  at  its  mean  intensity,  the  foregoing  expression  must 
be  multiplied  by  1.  Hence  we  have  for  the  value  of  the  attractive 

force  »•',  that  acts  to  cause  the  oscillations, 


The  general  expression,  (186),  gives  for  the  value  of  the  time  of 
the  oscillations  of  any  pendulum  under  the  action  of  an  attractive 
force,  g-, 

T=W—  . 

S 

If  we  call  the  length  of  the  common  pendulum,  whose  oscillations 
are  synchronous  with  those  of  the  bar  /,  we  have  by  substituting 
the  value  of  g,  from  (a),  and  squaring 


and  by  substituting  the  value  of  g-',  from  (6),  and  also  squaring 
T2_    *V/36 

m'fa  sin.  a  ' 
hence 

^R2/_    «V/36 

mf  ~m'fa  sin.  a  ' 


m      m'a  sin.  a 

whence  we  obtain 

m     I  R2a  sin.  a 


and  all  the  quantities  in  the  second  number  of  the  expression  may 
be  obtained  from  experiment ;  and  thus  the  ratio  between  the 
mass  of  the  earth,  and  that  used  in  the  experiment,  will  admit  .of 
calculation. 

*  See  Poisson,vol.  ii.  p.  42. 


300  or  COLLISION.  [Book  IV. 

CHAPTER  VI. 
OF  COLLISION. 

298.  The  simplest  mode  in  which  motion  can  be  communicated 
from  one  body  to  another  is  by  collision.     It  is  unnecessary  for 
us  to  inquire  whether  actual  contact  takes  place  in  this  case  be- 
tween the  particles  of  which  bodies  are  composed.    It  is  sufficient 
for  us  to  know  that  the  result  of  all  experiment  is  precisely  such 
as  would  happen,  were  the  contact  actually  to  occur. 

All  bodies  in  nature  are  more  or  less  compressible,  and  when 
they  have  been  compressed,  tend  in  agreateror  less  degree,  tore- 
cover  their  original  figure.  In  some,  this  tendency  is  extremely 
small ;  in  others  it  is  considerable  ;  it  is  styled  their  Elasticity. 
When  bodies  restore  themselves  to  their  original  figure,  after  be- 
ing compressed,  and  their  particles  in  restoring  the  figure,  return 
with  a  force  equal  to  that  by  which  the  compression  was  effected, 
they  are  said  to  be  perfectly  elastic.  If  they  did  not  yield  to  com- 

Sression,  they  would  be  hard,  and  wholly  devoid  of  elasticity, 
f  the  latter  class,  there  are  probably  no  bodies  in  nature  ;    but 
there  are  some  that  retain  any  figure  that  may  be  impressed  upon 
them,  having  little  or  no  tendency  to  restore  themselves  to  their 
original  shape. 

Gases  and  vapours  are,  within  certain  limits,  perfectly  elastic; 
bodies  of  other  classes  differ  materially  in  this  respect. 

299.  In  investigating  the  laws  of  collision,  we  consider  bodies  ei- 
ther as  perfectly  elastic,  or  as  wholly  devoid  of  elasticity ;  and  may 
thence  finally  conclude  what  would  occur  in  the  case  of  imperfect 
elasticity.     The  simplest  law  is  that  which  governs  the  collision 
of  non  elastic  bodies  ;  from  this,  too,  the  circumstances  of  the  col- 
lision of  perfectly  elastic  bodies,  may  be  directly  deduced.     The 
former  of  these  must,  in  consequence,  be  first  investigated  : 

Let  A  and  B  be  two  non-elastic  bodies,  homogeneous,  and  of  a 
spherical  figure  ;  and  let  their  centres  move  in  the  same  straight 
line,  in  such  a  manner,  that,  when  they  strike,  the  point  of  impact 
shall  be  in  the  line  that  joins  their  centres.  Let  a  and  6,  be  their 
respective  velocities,  which  if  in  the  same  direction,  will  have  the 
same,  if  in  contrary  directions,  opposite  algebraic  signs.  When 
they  strike  against  each  other,  each  will  yield  to  a  greater  or  less 
extent,  until  the  whole  action  have  taken  place,  and  the  time  in 
which  this  occurs  may  be  considered  as  inappreciably  small ;  in 
all  cases  it  is  in  fact  insensible.  So  soon  as  the  whole  action  has 
taken  place,  the  two  bodies  will  move  forward  with  equal  velocity, 


Book  IV.}  OF  COLLISION.  301 

for  their  is  no  force  to  act,  provided  they  be  non-elastic,  to  separate 
them.  Call  this  common  velocity,  v,  the  quantity  of  velocity  lost 
or  gained  by  A,  will  be  a  —  u  ;  and  that  lost  or  gained  by  B,  will 
be  6  —  v  ;  and  these  will  be  positive,  or  negative  quantities,  accord- 
ing to  the  conditions  of  the  problem. 

The  quantities  of  motion  the  bodies  respectively  lose  or  gain, 
will  be 

Ao  —  Ar,  and  B6  —  BTJ  ; 
hence  by  the  principle  of  D'  Alembert,  §  69, 

(Ao—  Aw)  +  (B6—  Bu)=0  ; 
whence  we  obtain  for  the  value  of  r, 


The  whole  quantity  of  motion  will  be, 

Ar-r-Bu  ;  (322 

and  the  respective  quantities  are,  Au,  and  Bt>. 

If  one  of  the  bodies,  A,  be  alone  in  motion,  and  B  at  rest,  6=0 
and 


If  the  bodies  move  in  opposite  directions,  a  and  6  will  have  con- 
trary signs  ;   and  if  a  be  positive, 

Aa—  B6 


If  the  bodies  be  equal,  and  the  velocities  equal,  but  in  contrary 
directions, 

v=0.  (325) 

If  the  bodies  have  equal  quantities  of  motion  in  opposite  directions 
Aa=B6,  (326) 

and 

A:  B::  6:0; 

in  which  case  we  again  have 

t=0.  (327) 

To  express  these  formulae  in  words  : 

When  two  non-elastic  homogeneous  bodies  of  a  spherical  form 
impinge  against  each  other,  at  a  point  situated  in  the  line  that 
joins  their  centres,  if  the  sum  of  their  quantities  of  motion  before 
impact,  be  divided  by  the  sum  of  their  masses,  the  quotient  is 
the  common  velocity  after  impact.  The  quantities  of  motion 
must  be  considered  as  having  the  same  or  contrary  signs,  accord- 
ing as  the  motions  are  in  the  same  or  contrary  directions. 

If  one  of  the  bodies  is  at  rest,  its  quantity  of  motion  is  to  be  di- 
vided by  the  sum  of  the  masses. 


308  or  COLLISION.  [Book  /F. 

If  th~  bodies  be  equal,  and  move  in  opposite  directions  with 
equal  velocities,  they  destroy  each  other's  motions,  and  both  come 
to  rest.  The  same  is  the  consequence  if  unequal  bodies  meet, 
with  equal  quantities  of  motion,  or  which  is  the  same  thing,  when 
their  velocities  in  opposite  directions  are  inversely  as  their 
masses. 

It  will  be  also  seen  from  inspection  of  the  formula  (321),  that 
the  change  of  motion  that  takes  place  in  each  of  the  bodies  is 
equal,  and  that  the  change  in  their  velocities  is  inversely  as  their 
masses. 

The  sum  of  their  motions,  estimated  in  the  same  direction,  is 
the  same  before  and  after  impact,  and  the  state  of  their  centre  of 
gravity,  whether  it  be  at  rest  or  in  motion,  is  not  changed  by 
their  mutual  action. 

If  one  of  the  bodies  be  infinitely  large  in  respect  to  the  other, 
and  at  rest,  then  the  common  velocity  becomes  infinitely  small, 
and  no  error  can  arise  in  taking 

t-=0. 

Such  is  the  case  in  all  obstacles  that  are  considered  as  immovea- 
ble,  which  are  so  only  in  consequence  of  their  great  magnitude, 
or  from  their  being  firmly  connected  with  the  surface  of  the  earth. 
If  their  surfaces  be  plane,  they  may  be  considered  as  portions  of 
spheres  of  infinite  magnitude.  Hence,  when  a  non-slastic  body 
strikes  against  a  plane  surface  in  the  direction  of  a  normal  to  it, 
it  will  be  brought  to  rest  upon  it ;  and  so,  as  we  may  consider 
curved  surfaces,  when  the  impact  takes  place  at  a  single  point,  as 
made  up  of  planes,  rest  will  again  take  place  whenever  a  spheri- 
cal and  homogeneous  non-elastic  body  strikes  in  the  direction  of  a 
normal  to  the  surface,  against  any  immoveable  obstacle. 

300.  If  a  spherical  body  strike  against  an  immoveable  plane 
surface  in  any  other  direction  than  t.hat  of  a  normal,  resolve  its 
moving  force  into  two  components,  one  of  which  is  in  the  direc- 
tion of  a  normal  to  the  surface,  and  the  other  coincides  with  the 
surface.  That  part  which  is  in  the  direction  of  the  normal,  will  be 
destroyed  by  the  resistance  of  the  surface  ;  the  other  part  will 
remain  acting  upon  the  body,  which  will,  therefore,  move  along 
the  surface  under  its  influence  ;  and  the  new  direction  will  be 
defined  by  its  being  in  the  plane,  passing  through  the  original  di- 
rection of  the  moving  body. 

301.  If  the   two  spherical  bodies  d/>  not  move  in  the  same 
straight  line,  but  have  the  directions  of  their  motions  inclined  to 
each  other,  the  bodies  will  go  on  together  in  the  direction  of  the 
resultant  of  their  respective  quantities  of  motion,  and  the  sum  of 
their  new  motions  will  be  represented  in  magnitude  by  this  re- 
sultant. 


Book 


OF  COLLISION. 


SOS 


The  determination  of  the  direction  and  quantity  of  motion  in  this 
case  may  be  effected  by  a  simple  geometric  construction,  which  is 
as  follows:  Draw  from  the  pointof  concourse  lines  to  represent  the 
magnitudes  and  directions  of  the  forces  whose  intensities  are  the 
respective  products  of  the  masses  of  the  two  bodies  into  their  ve- 
locities ;  call  these  lines  A  and  B.  Resolve  one  of  these,  B,  by 
the  construction  of  a  right  angled  parallelogram  into  two  others, 
one  of  which,  a,  is  parallel,  the  other,  6,  perpendicular  to  A :  pro- 
duce A  on  the  other  side  of  the  point  of  concourse,  until  the 
length  of  the  produced  part=A-fa;  produce  B  in  like  man- 
ner until  the  part  produced— b  :  on  the  two  lines  produced,  A-fa 
and  6,  construct  a  parallelogram  which  will  be  rightangled,  and 
draw  the  diagonal.  The  sum  of  the  motions  of  the  two  bodies 
after  impact,  will  be  represented  in  magnitude  and  direction  by 
this  diagonal ;  and  if  it  be  divided  into  two  parts,  proportioned  to 
the  masses  of  the  two  bodies,  these  will  represent  their  respective 
quantities  of  motion. 

302.  If  two  bodies,  moving  in  parallel  lines,  strike  against 
each  other,  the  circumstances  of  their  motion,  after  impact,  will 
differ  from  the  case  we  have  considered,  in  which  their  centres  of 
gravity  move  in  the  same  straight  line. 

Suppose,  first,  that  the  bodies  strike  in  such  a  manner  that  the 
direction  of  one  of  them  shall  be  a  common  normal  to  their  sur- 
faces. 

Let  KLM,  NOP,  be  sections  of  the  two  bodies,  in  a  plane 


304  or  COLLISION.  [Book  IV. 

passing  through  their  respective  centres  of  gravity,  and  the  line 
AC,  which  represents  the  direction  of  A  ;  this  direction  will  have 
for  its  distance,  ar,  from  the  centre  of  gravity  of  B,  the  line  BC. 
Let  a  be  the  velocity  that  remains  to  the  body  A,  after  impact. 
The  mass  B  being  acted  upon  by  a  force  whose  direction  does 
not  pass  through  its  centre  of  gravity,  will  derive  from  it  two  mo- 
tions, one  of  which  is  progressive,  the  other  rotary,  §  245,  around 
an  axis  passing  through  its  centre  of  gravity,  and  perpendicular 
to  the  plane  assumed  for  the  section  of  the  body.  Let  u  be  the 
velocity  of  the  progressive  motion,  t<>,  the  angular  velocity  of  the 
rotary  motion.  Let  a  and  6  represent  the  respective  motions  of 
A  and  B  before  impact. 

The  body  A  will  gain  or  lose  a  velocity  represented  by  a  —  a' 
the  body  B,  a  velocity  represented  by  6  —  M,  and  by  the  principle 
ofD'Alembert, 

A.  (a—  a')  -f-  B  (6—  w)  =0.  (328) 

The  body,  B,  will  revolve  with  a  force  equal  to  that  which  A 
loses  by  the  collision.  If  then  the  moment  of  inertia  of  the 
body,  B,  in  respect  to  the  axis  passing  through  its  centre  of  gra- 
vity, =  S,  we  shall  have  (241)  for  the  value  of  the  angular  velo- 
city, 

Ax  (a—  a') 
u=  -  ^j  --  -.  (329) 

For  the  same  reason  that  spherical  bodies,  §  297,  do  not  sepa- 

rate after  impact,  the  body,  A,  will  go  on  with  a  common  velocity 

with  the  point  C,  of  the  body  B.    This  velocity  is  u  —  xu,  hence, 

a'=u  —  xw.  (330) 

If  the  bodies  do  not  strike  in  a  point  that  is  in  the  common  nor- 
mal to  their  two  surfaces,  both  will  be  caused  to  revolve,  and  the 
rotary  motion  of  A,  may  be  estimated  upon  the  same  principles. 

303.  When  the  bodies  that  impinge  against  each  other  are 
elastic,  they  will  no  longer  go  on  together  with  a  common  velo- 
city, but  will  be  separated  by  the  action  of  their  elasticities.  If 
the  elasticity  be  perfect,  each  body  will  tend  to  restore  itself  to 
its  original  Torm,  with  a  force  equal  to  that  by  which  the  change 
of  figure  has  been  produced  ;  this  force  will  be  equal  to  the 
change  of  motion  it  would  undergo  were  it  non-elastic.  What- 
ever quantity  of  motion,  then,  a  body  loses  or  gains,  when  non- 
elastic,  will  be  exactly  doubled  if  it  be  perfectly  elastic.  On  this 
principle,  we  may  proceed  to  investigate  the  laws  of  the  collision 
of  perfectly  elastic  bodies. 

Supposing  the  bodies  to  be  spherical,  and  to  move  in  the  same 
straight  line,  using  the  same  notation  as  in  §  297  ;  their  common 
velocity,  if  non-elastic  would  be  (321) 


A+B 


Book  1V.~\  OF  COLLISION.  305 

the  body  A,  would,  in  this  case,  lose  or  gain  a  velocity  repre- 
sented by 


v  —  a 


and  it  will  lose  or  gain  as  much  more  on  account  of  its  perfect 
elasticity  ;  hence  its  actual  velocity  will  be 

a'=2«—  a;  (331) 

substituting  the  value  of  v,  we  have 
/  Ao—  B6\ 

=~ 


whence 

Aa—  B&+2B6 

a==      ~A+B  -  •  (332) 

or 

(A—  B)a-f2B6 

a'=      ~A+B~  ~  ;  <333> 

pursuing  the  same  course  of  reasoning  in  respect  to  B,  we  obtain 
B6—  A6+2  Ao 

"A+B  -  '  (334) 

or 

(B—  A)  6+2  Aa 

-A+S-  (33*) 

when  the  bodies  are  equal, 

a'  =6,  and  b'—a.  (336) 

The  relative  velocity  before  impact,  is  a  —  6,  after  impact, 
a'  —  6'  5  by  subtraction  of  the  two  last,  we  obtain 

a—  6  =—  a'+6',  (337) 

and 

a-r-a'=6+&',  (333) 

whence  we  have  for  the  sums  of  their  motions  before  and  after 
impact, 

Aa-|-B&=Aa'+B&'.  (339) 

Hence  we  may  infer,  that  — 

304.  When  two  perfectly  elastic  and  homogeneous  spherical 
bodies,  moving  in  the  same  straight  line,  impinge  against  each 
other,  they  separate  after  impact,  and  move  each  with  a  different 
velocity  ;  if  we  estimate  the  velocities  that  would  be  l8st  or 
gained  by  each,  if  non-elastic,  and  add  or  subtract  this,  to  or  from, 
the  common  velocity  they  would  have  upon  the  same  hypothesis, 
we  obtain  their  actual  velocities  after  impact. 

If  the  masses  of  the  bodies  be  equal,  their  velocities  are  inter- 
changed. Thus,  if  one  only  of  the  bodies  be  in  motion,  it  will 
come  to  reit  after  impact,  and  the  other  will  move  forward  with 
the  original  velocity  of  the  first  ;  if  they  move  in  opposite  direc- 

39 


306  OF  COLLISION.  [Book  IV. 

tions,  each  will  return  in  the  direction  whence  it  came,  with  the 
velocity  the  other  had  :  if  they  move  in  the  same  direction  with 
unequal  velocities,  the  more  swift  overtaking  the  more  slow,  that 
which  moved  most  rapidly  will  move  with  the  less  velocity  the 
other  had  before  impact,  and  vice  versa.  Whether  the  bodies 
be  equal  or  unequal,  the  difference  in  their  velocities,  or  what 
is  styled  their  Relative  Velocity,  will  be  the  same  in  amount  both 
before  and  after  impact;  but  they  will  have  contrary  signs. 

When  an  elastic  body  strikes  against  another  that  is  at  rest, 
and  larger  than  itself,  the  former  will  return  in  the  direction  in 
which  it  was  proceeding  before  impact;  and  if  the  latter  be  infi- 
nitely large  when  compared  with  the  former,  the  motion  of  re- 
turn or  reflection  will  have  the  same  velocity  with  that  of  impact. 
For  the  same  reason,  a  body  perfectly  elastic  will  be  reflected 
back  by  a  plain  surface,  in  a  direction  opposite  to  that  in  which 
it  came,  and  with  a  velocity  equal  to  that  it  originally  had. 

305.  If  the  impact  be  oblique,  the  reflection  will  not  take  place 
in  the  same  straight  line,  but  will  still  occur  ;  and  it  may  be  in- 
ferred, from  an  application  of  the  principle  of  "elasticity  to  §  298, 
that  the  directions  of  reflection  and  incidence  will  both  be  situa- 
ted in  the  same  plane  perpendicular  to  the  surface  of  impact,  and 
that  the  angles  of  incidence  and  reflection  will  be  equal  to  each 
other. 

306.  There  are  several  cases  where,  when  we  have  occasion  to 
estimate  mechanical  effect,  it  becomes  necessary  to  consider  the 
product  of  the  mass  by  the  square  of  the  velocity.  This  product  is 
called  the  Vis  Viva  of  the  body,  in  reference  to  which  it  is  esti- 
mated.    When  non-elastic  bodies  inpinge  against  each  other,  the 
sum  of  these  products  of  the  two  bodies  does  not  remain  constant  ; 
but  it  is  otherwise  with  bodies  that  are  perfectly  elastic. 

If  we  compare  the  sums  of  the  forces  estimated  in  the  above 
manner  in  the  case  of  non  elastic  bodies,  it  will  be  found  that  they 
have  diminished  ;  in  the  case  of  perfectly  elastic  bodies,  these 
products  remain  constant,  as  may  be  readily  shown. 

Taking  the  formula,  (339), 

Aa+B6=Aa'+B6', 
and 

A(»_a')=B(&-6'); 
then  by  (338), 


".  (340) 


multiplying  these  two  equations,  we  have 
whence  we  obtain 


Book  IV.  1  OF  COLLISION.  307 

307.  When  a  small  elastic  body  in  motion  strikes  against  a 
greater  elastic  body  at  rest,  it  has  been  stated  that  the  fortnet 
would  be  reflected  ;  the  larger  will  be  set  in  motion,  and  will 
proceed  forward  in  the  direction  in  whfch  the  smaller  was  pro- 
ceeding before  impact.  This  will  be  readily  seen  by  consider- 
ing, that  were  the  bodies  not  elastic,  the  whole  motion  would  be 
distributed  among  them  in  proportion  to  their  masses  ;  hence, 
when  we  add  as  much  more  motion  to  that  the  larger  body  would 
receive  under  this  hypothesis,  the  sum  will  he  more  than  the  origi- 
nal quantity  of  motion  in  the  smaller  body.  The  whole  quan- 
tity of  motion  in  the  two  bodies  is  not  on  this  account  increased, 
for  the  smaller  body  is  reflected  with  a  quantity  of  motion  ex- 
actly equal  to  the  increased  amount  communicated  to  the  greater 
body,  and  the  algebraic  sum  of  the  motions  estimated  in  one  di- 
rection is  constant. 

When  several  elastic  bodies,  increasing  in  magnitude,  are  ar- 
ranged in  the  same  line,  touching  each  other,  and  one  smaller 
than  the  least  of  them  strikes  against  it,  each  will  in  turn  be  re- 
flected, and  communicate  to  the  succeeding  one  a  greater  quantity 
of  motion  than  itself  has.  This  increase  of  motion  is  greatest 
when  the  masses  of  the  bodies  increase  in  geometric  progression. 

Let  there  be  three  elastic  homogeneous  bodies  of  spherical 
figure,  A,  B  and  C,  of  which  A  is  in  motion,  and  strikes  against  B, 
in  a  line  passing  through  the  centres  of  the  three  bodies  ;  the  ve- 
locity communicated  to  B  will  be,  (334), 


A+5 

and  to  C,  calculated  also  from  (334), 

4ABa  4Art 


(A+B)X(B+C)-    A     I)(B+C) 


4Aa  (342) 


AC 
A+B-f-g— i-C  ; 

if  the  magnitudes  of  A  and  C  are  given,  the  last  expression  will 
obviously  be  a  maximum,  when 

B3-AC, 

or  when  A,  B  and  C,  are  in  geometric  progression,  and  the  same 
may  be  proved  of  any  number  of  elastic  bodies  whatsoever. 
308.  When  the  bodies  before  collision  move  in  parallel  lines, 
or  when  they  meet  in  lines  making  an  angle  with  each  other, 
the  principles  of  §  299  may  be  made  use  of  to  estimate  the  quan- 
tities of  motion  that  would  be  lost  or  gained  by  each,  if  non  elas- 


308  OF  COLLISION.  [Book  IV. 

tic,  and  the  change  that  is  produced  in  the  direction  of  the  mo- 
tion. Knowing,  then,  that  both  the  change  in  the  direction,  and 
in  the  quantity  of  motion,  is  doubled  by  perfect  elasticity,  the 
manner  in  which  the  bodies  will  move  in  the  first  case,  and  the 
direction  and  intensity  in  the  second,  subsequent  to  collision, 
may  be  investigated.  In  these  cases,  the  general  formulae  of  the 
composition  and  resolution  of  forces  will  meet  every  possible  in- 
stance. 

309.  If  we  examine  into  the  state  of  the  centre  of  gravity,  we 
shall  find  that  whether  it  be  in  rest  or  in  motion  before  the  colli- 
sion of  the  bodies,  that  state  of  rest  or  motion  will  not  be  affected 
by  their  impact  ;  the  same  has  been  found  to  be  the  case  in  the 
collision  of  non-elastic  bodies  ;  and  this  is  a  constant  law  in  what- 
ever manner  the  bodies  act  upon  each  other,  and  whatever  be 
their  respective  natures. 

310.  When  the  elasticity  of  bodies  that  impinge  against  each 
other  is  not  perfect,  the  quantity   of  motion  lost  or  gained  by 
each,  in  consequence  of  their  elasticities,  will  be  less  than  in  the 
case  of  perfect  elasticity. 

Let  p  be  the  relation  of  the  force  of  elasticity  to  the  compress- 
ing force,  the  formulae,  (333)  and  (335),  will  become 

B(o—  6) 
a<=a_(l+p) 
and 


If  we  estimate  the  loss  of  vis  inta,  which  takes  place  in  conse- 
quence of  collision,  we  shall  find  it  to  be 
AB(o—  6)a 

A+B      *  (343) 

when  the  bodies  are  perfectly  elastic,  p=l,  and  this  expression 
=0  ;  and  when  non-elastic,  />=0,  and  the  expression  is  a  maxi- 
mum. 


BOOK   V. 

OF   THE  EQUILIBRIUM  OF  FLUIDS. 

CHAPTER  I. 

GENERAL  CHARACTERS  01-  FLUID  BODIES. 

311.  Fluids  are  distinguished  from  solid  bodies,  §  79,  by  the 
greater  degree  of  ease  with  which  their  particles  may  be  separa- 
ted. It  is  the  distinctive  and  characteristic  property  of  fluids, 
that  these  particles  may  be  moved  among  each  other  by  the 
smallest  possible  force.  The  physical  agent  that  causes  fluidity  is 
heat,  existing  in  a  latent  state. 

When  the  repulsive  force,  exerted  by  heat,  exactly  balances  the 
attractioi\  of  aggregation  that  exists  among  the  particles,  the  body 
is  a  liquid  ;  when  the  force  of  heat  preponderates,  the  body  has 
a  tendency  to  expand  itself;  this  tendency  is  opposed,  and  may 
be  overcome  by  pressure,  and  hence  the  body  will  occupy  spaces 
of  different  extents,  that  will  depend  upon  the  intensity  of  the 
compressing  force  :  such  a  fluid  is  said  to  be  elastic. 

We  know  of  no  perfect  liquids  in  nature ;  in  them  all  the  force 
of  the  attraction  of  aggregation  slightly  exceeds  the  repulsive 
force  of  heat,  as  is  manifested  by  small  portions  of  liquids  ar- 
ranging themselves  in  the  form  of  spherical  drops,  and  by  their 
particles  resisting  forces  that  tend  to  separate  them,  although  with 
a  feeble  intensity  ;  this  resistance  is  called  Viscidity. 

312.  Liquids,  in  examining  the  theory  of  their  equilibrium 
and  motion,  are  considered  as  incompressible,  and  consequently 
wholly  devoid  of  elasticity.  A  want  of  this  property  was,  for  a 
longtime,  considered  to  be  essential  to  liquids.  It  might,  how- 
ever, have  been  inferred,  that  as  liquids  are  capable  of  contracting 
and  expanding  with  changes  of  temperature,  they  were  not  ab- 
solutely incompressible;  and  that  a  mechanical  agent  of  intensi- 
ty equal  to  heat,  might  cause  them  to  change  their  bulk. 


310  GENERAL  CHARACTERS  [Book  V. 

This  inference  has  been  confirmed  by  the  experiments  of  Can- 
ton, and  more  recently  by  those  of  Perkins  and  Oersted,  who 
have  shown,  that  the  bulk  of  water  may  be  diminished  by  pres- 
sure. This  diminution  in  bulk  is  about  .000048  for  a  pressure 
equivalent  to  15  Ibs.  upon  a  square  inch  of  surface  ;  an  amount 
that,  as  will  be  shown  hereafter,  constitutes  that  unit  of  pressure 
which  is  called  an  Atmosphere. 

313.  To  determine  the  nature  and  intensity  of  the  forces  that 
act  upon  any  given  particle  of  a  fluid  mass,  we  may,  in  conse- 
quence of  the  very  small  size  of  the  particles,  and  the  apparent 
continuity  of  the  mass,  ascribe  to  these  particles  any  figure  that 
we  think  proper,  or  which  is  most  convenient  foe  the  purposes  of 
our  investigation. 

If  we  take  then  a  given  particle  of  the  fluid,  and  refer  its  posi- 
tion to  three  rectangular  co-ordinates,  X,  Y  and  Z,  we  may  con- 
sider the  particle  as  occupying  a  space  extended  in  three  dimen- 
sions, and  having  the  figure  of  a  parallelepiped,  whose  dimensions 
are  dr,  dy  and  dz  ;  and  may  assume  for  its  volume  the  product 
of  its  three  dimensions.  If  we  call  its  density  s,  its  mass  will  be 
adxdydz.  (344) 

The  forces  that  act  upon  this  particle,  must  be  either  inherent 
in  the  particle  itself,  and  intrinsic  ;  or  must  be  exercised  m  the 
form  of  pressure  by  the  surrounding  particles,  and  therefore  ex- 
trinsic. 

The  intrinsic  forces  may,  §  17,  be  resolved  into  thtfe,  acting 
at  right  angles  to  each  other,  in  the  directions  of  rfie  Aree  axes, 
X,  Y,  Z  ;  if  we  call  these  components  P,  Q  and  K,  the  forces 
that  solicit  the  particle  will  be  found  by  multiplying  /hese  compo- 
nents respectively  by  its  mass,  or  they  will  be 
P*  dx  dy  dz,  > 

Qsdxdydz,  }  (345) 

.>•/ jd(:Kj  ye  •>,{)  *  R*  dx  dy  dz.  I 

The  extrinsic  forces  are  pressures,  awl  may  be  represented  in 
each  case  by  taking  the  surface  dh  on  which  the  pressure  is  ex- 
erted, and  multiplying  it  by  the  keight  of  a  column  of  homoge- 
neous matter,  whose  weight  is  equal  to  this  pressure,  and  by  the 
density  of  this  column.  If  we /suppose  the  density  of  the  column 
to  be  1,  and  call  its  height  p,  the  pressure  will  be 

pdh  (346) 

This  pressure  must  be  exercised  in  the  direction  of  a  normal  to 
the  surface  dk :  for  were  the  force  oblique  ;  of  its  two  rectangular 
components,  one  in  the  direction  of  the  surface  itself,  the  other  in 
that  of  its  normal ;  the  latter  alone  could  produce  any  effect.  And 
it  is  therefore  obvious,  that  all  the  forces  that  can  act  upon  a  given 
surface  of  a  fluid  particle,  must  be  finally  reducible  to  two,  act- 


Book   V.~\  OF  FLUID  BODIES.  311 

ing  in  opposition  to  each  other,  in  the  direction  of  its  normal  ; 
these  forces  may  be  called 

pdk,  a.ndp'dk.  (347) 

And  in  the  case  of  equilibrium,  these  pressures  must  be  equal, 
and 

P=P'-  (348) 

If  the  surface  pressed  change  its  direction,  without  having  its 
position  in  the  mass  of  fluid  changed,  the  value  of  p  will  remain 
constant.  For  :  suppose  the  surface  to  turn  around  one  of  its 
edges,  until  it  make  with  its  original  position  the  angle  t  :  The 
magnitude  of  the  surface  that  is  pressed  by  the  column,  the  latter 
remaining  unchanged  in  area  and  volume,  will  be 

dk 

cos.  i  ' 

the  length  of  the  perpendicular  height  of  the  column  in  its  new  po- 
sition, from  the  surface,  will  be 

p  cos.  i  ; 

the  product  of  these  two  quantities  is  the  amount  of  the  pressure, 
or 

dk  '     n 

—{  .  p  cos.  t=dkp. 

But  in  different  parts  of  the  same  mass,  the  quantity  p  will  vary, 
and  will  depend  upon,  or  be  a  function  of  X,  Y  and  Z. 

If  then  we  take  another  surface  contiguous  to  the  first,  the 
column  that  measures  the  pressure  upon  it  will  have  for  its  alti- 

tude, 


and  the  pressure  will  be,  if  the  surface  have  the  same  area  as  the 
first, 


To  apply  these  principles  to  the  case  of  the  elementary  parallele- 
piped :  p  is,  as  has  been  stated,  a  function  of  the  three  co-ordi- 
nates ;  we  may,  therefore,  assume  for  the  value  of  its  differential, 


and  the  three  co-efficients,  L,  M  and  N,  will  be  the  differentials 
of  jp,  in  the  respective  directions  of  X,  Y  and  Z. 

The  face  whose  surface  is  dij  dz  will  be  pressed  in  the  direction 
of  a  normal,  which  direction  will  be  the  same  as  that  of  #,  by  a 
force  represented  by 

p  dy  dz  ; 

the  opposite  and  equal  face  of  the  parallelepiped  will  be  pressed 
by  a  force  in  the  opposite  direction,  whose  measure  is  a  column 
or  prism,  whose  base  is  dy  dz,  and  altitude  p+L  dx  ;  this  force 
will  therefore  be 

(p-f  L  dx]dy  dz  ; 


312  GENERAL  CHARACTERS  [Book  V. 

and  upon  the  remaining  four  surfaces,  the  pressures  will  be 

p  dy  dx, 
(p+Kz)dy  dx, 

p  dz  dx, 

(p+M  dy)dz  dx. 

These  six  forces,  acting  in  opposite  directions  by  pairs*  may 
be  resolved  into  three,  which  are 

L  dx  dy  dz,  \ 

M  dx  dy  dz,  }  (349) 

N  dx  dy  dz.  ) 

These  three  rectangular  forces  will  be  the  resultants  of  all  the  ex- 
trinsic forces  that  act  upon  the  particle.     If  with  these  be  com- 
bined the  rectangular  resultants  of  the  intrinsic  forces,  (345),  we 
have,  for  the  rectangular  resultants  of  all  the  forces  that  act, 
(Pa—  L)dxdydz,  } 
(Q«—  M)dx  dy  dz,  }  (350) 


314.  The  nature  and  intensity  of  the  forces  that  act  upon  any 
given  particle  of  the  fluid,  being  obtained  in  the  preceding  ex- 
pressions, we  may  proceed  to  investigate  the  conditions  of  equi- 
librium that  exist  among  them. 

As  every  particle  of  a  fluid  mass  has,  by  the  definition  of  that 
class  of  bodies,  perfect  freedom  of  motion,  except  such  resistances 
as  may  arise  from  the  pressure  of  the  surrounding  particles;  every 
particle  must  in  consequence,  when  the  mass  is  in  equilibrio,  be 
also  in  equilibrio  under  the  forces  that  act  upon  it.  The  three 
rectangular  resultants  of  the  forces,  (350),  must  therefore  be  res- 
pectively =0.  Hence,  in  the  expressions,  (350),  we  have 

L=P«,  M=Q«,  N=R«; 

if  we  multiply  these  equations  respectively  by  dx,  dy,  dz,  and  add 
the  products  together,  we  obtain 

dp=s(P  rf#+Q  <%+R  dz),  (351) 

which  is  the  equation  that  gives  the  conditions  of  equilibrium. 

315.  The  resistance  to  the  motion  of  the  particles  of  fluids,  is 
restricted  to  the  pressure  of  the  surrounding  mass  ;  while  in  solids, 
not  only  does  this  resistance  act,  but  one  of  far  greater  intensity, 
namely,  the  attraction  of  aggregation  that  exists  among  their  par- 
ticles.    The  above  condition  of  equilibrium  therefore  exists  in 
solids  as  well  as  in  fluids;  that   is  to  say,  that  if  it  obtain,  the 
mass  will  be  in  equilibrio,  but  we  cannot  infer,  that  if  it  do  not 
obtain,  the  mass  will  therefore  cease  to  be  in  equilibrio. 

It  results  from  the  nature  of  fluid  bodies,  whose  particles  are 
free  to  move,  that  if  any  one  of  them  be  set  in  motion,  all  the 


Book   V.~\  OF  FLUID  BODIES.  313 

rest  must  be  set  in  motion  also  ;  and  that  the  application  of  any 
force  to  any  one  of  the  particles,  in  addition  to  those  whose  re- 
sultants are  given  in  the  equation  (351),  will  affect  the  equilibri- 
um of  all  the  rest;  therefore  a  pressure  applied  to  any  point  of  a 
fluid  mass,  will  be  propagated  in  all  directions,  and  influence 
every  particle  of  which  the  mass  is  made  up.  A  pressure  ap- 
plied to  any  point  of  a  solid  body,  would,  in  like  manner,  be 
propagated  in  all  directions,  were  it  not  that  it  is  counteracted  by 
the  attraction  of  aggregation.  Hence,  the  property  usually  as- 
sumed as  the  basis  of  the  mechanics  of  fluid  bodies,  namely,  that 
of  propagating  pressure  equally  in  all  directions,  is  not  distinc- 
tive ;  and  solids,  in  which  the  attraction  of  aggregation  is  weak, 
or  in  which  it  has  been  overcome  by  mechanical  action,  will  press 
to  a  certain  extent  in  conformity  with  the  laws  that  govern  the 
pressure  of  fluids.  This  is  found  to  be  the  case  in  masses  of  sand 
and  loose  earth,  that  often  produce  mechanical  effects  similar,  al- 
though not  absolutely  identical  with  those  produced  by  fluids. 
In  consequence  of  this  mode  of  action,  the  investigation  in  §  196, 
of  the  strength  of  a  wall  that  will  support  the  pressure  of  earth, 
does  not  always  give  sure  results. 


40 


314  EQUILIBRIUM    OF  [Book    V. 

% 

CHAPTER  II. 

OF  THE  EQUILIBRIUM  OF  GRAVITATING  LIQUIDS. 

316.  All  fluids  upon  or  near  the  surface  of  the  earth,  are  in- 
fluenced by  the  attraction  of  gravitation  ;  in  liquids  thus  situated, 
no  other  cause  exists  for  the  mutual  action  of  their  particles  upon 
each  other.     This  force  is  always  exerted  in  the  direction  of  a 
line  tending  to  the  centre  of  the  earth,  at  which  point  all  the  di- 
rections of  the  force  of  gravity  at  the  surface  would  meet,  were 
the  earth  a  perfect  sphere.     The  direction  then,  at  any  place,  is 
a  vertical  line. 

If  we  suppose  that  this  direction  coincides  with  the  axis  *,  the 
quantities  P,  and  Q,  in  the  direction  of  the  other  axes,  become 
in  liquids =0.  R,  alone  remains,  and  if  we  consider  this  as 
equal  to  unity,  the  equation  (351)  becomes 

dp=*d2;  (352) 

and  integrating, 

p=«(A+Z);  * 

the  force  p,  which  measures  the  pressure  upon  any  point  of  a  gra- 
vitating liquid,  will,  therefore,  be  a  function  of  the  variable  quan- 
tity 2,  or  will  depend  upon  its  vertical  depth  beneath  the  surface 
of  the  fluid.  Hence,  every  point  situated  at  the  same  distance 
below  the  surface  of  a  mass  of  fluid,  will  undergo  an  equal  pres- 
sure. „ 

At  the  surface  of  a  gravitating  fluid,  the  quantity  dp,  which  de- 
pends upon  the  depth  of  the  fluid,  is  =  0  ;  hence,  the  equation 
(351)  of  the  surface,  when  in  equilibrio,  becomes 

»(Pdx+Qdi/+Rd*)=0; 
if  we  take  the  density  *=1, 

P  dx+  Q  efy+R  dz=0  ;  (353) 

and  in  the  case  of  a  liquid, 

dz=0.  (354) 

317.  When  a  gravitating  liquid  is  placed  in  an  open  vessel,  its 
particles  move  in  consequence  of  the  fundamental  property  of 
fluids,  until  they  reach  a  state  in  which  the  conditions  of  equili- 
brium are  satisfied.     It  will  then  come  to  rest,  and  assume  an 
uniform  and  constant  surface.     The  last  equation  (354)  shows 
that  every  point  in  this  surface  will  be  at  an  equal  distance  from 
the  plane  to  which  the  axis  z  is  perpendicular  ;  the  surface  will, 
therefore,  be  plane,  and  at  right  angles  to  the  direction  of  the 
force  of  gravity.     The  last  part  of  the  proposition  is  also  true, 


Book  T7".]  GRAVITATING  LIQUIDS.  315 

when  the  extent  of  the  surface  becomes  so  great  that  the  direc- 
tions of  gravity  can  no  longer  be  considered  as  parallel  ;  but  in 
this  case,  thesurface  itself  becomes  curved,  and  beingevery  where 
at  right  angles  to  the  force  of  gravity,  is  parallel  to  the  mean  sur- 
face of  the  terrestrial  spheroid.  Such  a  surface  is  said  to  be 
level,  and  is  spontaneously  assumed  by  all  stagnant  fluids  upon 
the  surface  of  the  earth  ;  and  in  masses  that  are  agitated  by  ex- 
trinsic forces,  as  the  ocean,  which  is  in  constant  motion  under 
the  action  of  the  winds,  and  the  causes  that  produce  the  tides, 
the  mean  surface  is  level. 

That  the  surface  of  a  liquid,  when  in  a  state  of  equilibrium  un- 
der the  action  of  any  force  whatsoever,  must  be  perpendicular  to 
the  direction  of  that  force,  may  be  shown  in  another  manner. 
For  if  we  assume  that  the  surface  has  not  this  direction,  the  force 
that  acts  on  any  particle,  may  be  resolved  into  two,  one  perpen- 
dicular, the  other  parallel  to  the  surface  ;  the  latter  then  will  not 
be  counteracted  by  fluid  pressure,  and  the  particle  will  move  un- 
der its  influence,  which  is  contrary  to  the  hypothesis  of  equili- 
brium. This  mode  of  considering  the  subject  shows  in  addition, 
what  does  not  appear  from  the  equation  of  equilibrium,  namely, 
that  the  force  that  causes  a  fluid  to  assume  a  definite  surface,  must 
be  directed  from  without  the  mass  towards  its  interior. 

318.  The  art  of  levelling  consists  in  drawing  a  line,  which 
shall  every  where  intersect  the  direction  of  gravity  at  right  an- 
gles.    Such  a  line  is  in  fact  a  curve,   although,  it  may  be  con- 
sidered as  straight,  within   narrow  limits.     Our  instruments  do 
not  furnish  the  means  of  drawing  the  level  curve,  and  we  are, 
therefore,  compelled  to  content  ourselves  with  obtaining  tangents 
to  it,  at  the  several  points  of  observation.    If  these  tangents  be  of 
no  great  length,  and  if  they  be  equally  produced  each  way  from 
the  point  of  contact  with  the  curve,  they  form  a  polygon,  that 
may,  without  sensible  error,  be  considered  as  identical  with  the 
curve  itself. 

The  levelling  instruments,  in  most  frequent  use,  are: 

The  Water  Level,  the  Spirit  Level,  and  the  Mason's  Level. 

319.  The  water  level  is  formed  of  a  tube  of  glass,  bent  twice 
at  right  angles.      If  a  fluid  be  placed  in  this  tube,  its  surfaces  will 
rise  in  each  branch  to  the  same  height  above  the  mean  surface, 
as  will  be  shown  in  a  succeeding  section.     If  the  sight  be  directed 
over  these  two  surfaces,  the  line  of  sight  will  be  of  course  a  tan- 
gent to  the  level  curve. 

320.  The  essential  part  of  a   spirit  level  consists  of  a  cylin- 
drical tube  of  glass,    containing   a    portion   of  spirits   of  wine 
that  nearly  fills  It ;  this  tube  is  hermetically  sealed  at  each  end. 


316  EQUILIBRIUM  OF  [Book    V, 

The  space  in  the  tube  that  is  not  occupied  by  the  spirit,  contains 
air,  which  appears  in  the  form  of  a  bubble.  The  air  being  much 
lighter  than  the  spirit,  the  bubble  will  always  occupy  the  highest 
part  of  the  tube  ;  and  when  the  tube  is  truly  horizontal,  the  bub- 
ble will  come  to  rest  at  a  distance  exactly  equal  from  each  end. 
Any  deviation  from  a  horizontal  position,  will  be  marked  by  an 
approach  of  the  bubble  to  the  end  of  the  tube  that  is  most  ele- 
vated. 

In  order  to  render  the  indications  of  the  level  more  precise,  it 
has  been  improved  by  grinding  the  interior  of  the  tube  into  the 
form  of  a  portion  of  a  circular  ring  of  large  radius  ;  and  the  axis 
of  the  tube  is  in  consequence  no  longer  a  straight  line,  but  an  arc 
of  a  circle.  If  the  radius  of  this  circle  be  known  in  some  con- 
ventional unit  of  measure,  and  divisions  of  the  same  unit  be 
marked  upon  the  tube,  the  deviation  of  the  bubble  from  the  ex- 
act middle  of  the  tube  may  be  measured  by  these  divisions,  and 
hence  the  inclination  of  the  chord  or  line  that  joins  the  extreme 
points  of  the  axis,  estimated. 

This  level  is  usually  mounted  upon  a  tripod  stand,  that  bears 
two  plates  that  are  ground  to  fit  each  other  ;  one  of  these  is  at 
rest,  the  other  moves  upon  it,  around  a  vertical  axis,  and  carries 
with  it  the  tube.  The  conditions  on  which  accuracy  depends  in 
this  part  of  the  arrangement,  are,  that  the  axis  of  the  plates'  motion 
shall  be  truly  vertical ;  and  that  the  axis  of  the  tube  shall  be  pa- 
rallel to  the  surfaces  of  the  plates,  and  of  course  perpendicular  to 
the  vertical  axis  of  motion. 

Whether  both  of  these  conditions  be  fulfilled,  may  be  ascer- 
tained by  one  and  the  same  operation,  and  the  instrument  ad- 
justed, if  necessary.  The  direction  of  the  axis  of  the  motion  of 
the  plate,  is  capable  of  change  in  respect  to  the  stand,  by  means 
of  four  screws,  called  the  levelling  screws  of  the  instrument. 
Such,  at  least,  is  the  more  usual  number,  and  to  assume  that  num- 
ber as  adopted,  will  best  suit  the  purpose  of  explanation ;  al- 
though three  screws  would  probably  afford  a  more  convenient 
adjustment,  as  may  be  inferred  from  §  112.  The  tube  being  brought 
directly  over  two  of  the  screws,  they  are  moved  in  opposite  di- 
rections until  the  bubble  exactly  occupies  the  middle  of  the  tube  ; 
the  axis  of  the  tube  is,  therefore,  level.  In  order  to  determine  whe- 
ther it  be  parallel  to  the  plates,  and  whether  the  axis  of  the  latter  be 
vertical,  the  moveable  plate  is  turned  around  its  axis  through  the 
half  of  a  complete  revolution  ;  the  position  of  the  levellingtubeis, 
therefore,  reversed.  If  in  this  new  position,  the  bubble  still  oc- 
cupy the  middle  of  the  tube,  the  adjustment  is  complete.  If, 
however,  the  bubble  have  approached  to  either  end,  one  half  of 
its  change  of  position  is  obviously  owing  to  a  want  of  parallelism 
in  the  motions  of  the  plate  and  the  tube  ;  the  other  half,  to  the 


Book   J7".]  GRAVITATING  LIQUIDS.  317   , 

axis  not  being  vertical.  In  order  to  correct  the  latter  error,  the 
levelling  screws  are  again  moved,  until  the  bubble  have  passed 
back  through  about  half  the  space  thit  marks  its  change  of  posi- 
tion. To  enable  us  to  remedy  the  former  error,  the  glass  tube 
that  contains  the  spirits  of  wine  is  enclosed  in  another  tube  of 
brass,  open  at  its  upper  surface,  to  permit  the  bubble  to  be  seen, 
and  the  divisions  of  the  scale  to  be  read.  This  outer  tube  is  at- 
tached to  the  bar  that  supports  it,  at  one  end,  by  a  hinge,  whose 
axis  is  horizontal,  and  at  the  other  by  a  vertical  screw.  The 
bubble  having  been  moved  back  by  the  means  just  described, 
through  half  the  distance  it  before  passed  through,  in  conse- 
quence of  the  error  in  adjustment,  is  brought  exactly  to  the  mid- 
dle of  the  tube  by  the  last  named  screw.  The  instrument  being 
then  turned  around  until  it  reach  its  original  position,  the  bubble 
is  again  inspected;  if  it  now  come  to  rest  exactly  in  the  middle 
of  the  tube,  the  adjustment  is  complete  ;  if  not,  the  preceding 
operations  must  be  repeated,  until  the  bubble  stand  exactly  in  the 
middle,  in  both  positions  of  the  tube. 

In  order  to  direct  the  vision  to  the  signals  placed  at  the  sta- 
tions whose  relative  level  is  to  be  determined,  sights  must  be 
adapted  to  the  tube  ;   that  which  has  superseded  all  others,  is  the 
telescope  ;  and  the  level,  instead  of  being  mounted  as  we  have 
assumed,  for  the  sake  of  more  easy  illustration,  upon  a  simple 
bar,  is  attached  by  the  hinge  and  vertical  screw  to  the  tube  of 
the  telescope.     The  telescope  itself  is  mounted  upon  the  horizon- 
tal bar  that  turns  around  with  the  moveable  plate.     For  this  pur- 
pose, the  latter  is  furnished  with  two  vertical  supports;  these 
have  the  form  of  rods,  divided  at  their  upper  end  into  two  in- 
clined branches,  and  thus  having  the  form  of  the  letter  Y,  whence 
this  part  of  the  apparatus  derives  its  name.     One  of  these  Ys  has 
a  motion  in  a  vertical  direction,  by  means  of  a  screw  placed  be- 
neath it.     The  telescope  has  upon  its  tube  two  collars,  at  an  ap- 
propriate distance  from  each  other,  that  are  accurately  turned, 
and  rest  upon  the  Ys.     It  is  obvious,  that  the  axis  of  the  tele- 
scope, or  more  properly,  the  line  that  joins  the  centres  of  the  two 
collars,  must  be  rendered  parallel  to  the  axis  of  the  spirit  level. 
To  ascertain  if  this  be  the  case,  and  to  correct  it  if  it  be  not :  the 
previous   adjustment  having  been   completed,    and   the    bubble 
brought  exactly  to  the  middle  of  the  tube,  the  telescope  is  lifted 
from  the  Ys,  and  again  replaced  in  an  inverted  position  ;  that  is 
to  say,  each  of  the  collars  is  made  to  rest  in  the  Y  that  had  before 
borne  the  other.      If  in  this  inverted   position  the  bubble  still 
occupy  the  middle  of  the  tube,  this  adjustment  is  complete  ;  but 
if  it  do  not,  one  half  of  the  error  is  due  to  the  position  of  the  pa- 
rallel plates,  the  other  half  to  the  position  of  the  Ys.      In  order, 
therefore,  to  effect  this  adjustment,  the  bubble  is  brought  back  to 


318  EQUILIBRIUM  OF  [Book   Jr. 

the  middle  of  the  tube,  partly  by  the  levelling  screws  of  the  in- 
strument, and  partly  by  the  screw  that  has  been  described  as  giv- 
ing a  vertical  motion  to  one  of  the  Ys. 

A  telescope  having  a  field  of  view  of  a  definite  extent,  it  is 
necessary  that  some  line  should  be  defined  within  it,  to  point  the 
direction  of  the  vision  to  the  signals.  This  is  effected  in  the 
levelling  instrument,  by  means  of  the  intersection  of  two  wires, 
the  one  vertical,  and  the  other  horizontal.  These  are  placed  in 
the  common  focus  of  the  two  lenses  of  the  telescope,  and  are,  in 
consequence  of  one  of  the  optical  properties  of  that  instrument, 
as  distinctly  visible  as  the  signal.  The  line  passing  through  the 
eye,  and  the  intersection  of  these  wires,  is  called  the  Line  of 
Collimation.  In  order  to  adjust  this  ;  the  telescope,  after  the 
instrument  has  received  the  previous  adjustment,  is  directed 
towards  a  moveable  vane,  on  which  a  horizontal  line  is  drawn  ; 
the  vane  is  raised  or  lowered,  until  this  line  apparently  coincides 
with  the  horizontal  wire,  in  the  focus  of  the  telescope.  The  tele- 
scope is  next  turned  half  round,  in  the  Ys,  on  its  own  horizontal 
axis;  if  the  line  on  the  vane  correspond  exactly  with  the  hori- 
zontal wire,  the  instrument  requires  no  farther  adjustment;  but 
if  these  lines  do  not  coincide,  half  the  error  is  evidently  due  to 
the  positron  of  the  vane  ;  the  other  half  to  that  of  the  horizontal 
wire  in  the  focus  of  the  telescope.  The  correction  is  again  double  ; 
the  vane  is  moved  through  half  the  space  that  intervenes  between 
the  upper  and  lower  visible  position  of  the  telescope  ;  and  the 
horizontal  wire  of  the  telescope,  is  moved  through  the  other  half, 
by  means  of  screws,  adapted  for  the  purpose,  to  the  tube  of  the 
telescope. 

We  determine  in  these  adjustments,  and  in  the  subsequent  use 
of  the  instrument,  the  proper  position  of  the  bubble,  by  observing 
the  place  occupied  by  its  two  extremities  ;  for  the  bubble  has  no 
direct  mode  of  showing  the  position  of  its  exact  middle.  The 
length  of  the  bubble  is  constantly  varying  with  changes  of  tempe- 
rature, for  the  spirits  of  wine  expand  and  contract,  when  heated 
or  cooled,  and  the  air  being  elastic,  accommodates  itself  to  this 
change  of  bulk.  Exposure  to  the  sun  affects  the  temperature, 
and  his  rays  sometimes  fall  obliquely  and  partially.  In  the  latter 
case,  the  indications  of  the  spirit  level,  are  uncertain,  and  the 
bubble  will  move  from  its  position,  when  the  instrument  itself  is 
perfectly  at  rest. 

321.  In  using  the  level,  it  may  be  placed  at  one  of  the  poinfs, 
whose  difference  of  elevation  is  to  be  determined  ;  when  the  in- 
strument has  been  levelled  by  bringing  the  bubble  to  the  middle 
of  the  tube,  by  the  motion  of  the  levelling  screws,  and  it  has  been 
ascertained  that  the  bubble  will  continue  in  that  position  while  it 


Book   VJ\  GRAVITATING  LIQUIDS.  319 

moves  around  the  vertical  axis,  the  telescope  is  directed  to  the  other 
point ;  at  this  a  staff  is  set  up  in  a  vertical  position,  on  which  a  vane 
slides.  The  vane  has  a  horizontal  line  marked  upon  it,  and  an  as- 
sistant slides  the  vane  upon  the  staff,  until  this  horizontal  line  ap- 
pears to  coincide  with  the  horizontal  wire  of  the  telescope.  The 
difference  between  the  height  of  the  eye-piece  of  the  instrument, 
and  that  of  the  horizontal  line  upon  the  vane  measured  from  the 
points  on  which  they  respectively  stand,  is  the  difference  of  level. 
This  determination  wiM  require  corrections,  unless  the  dis- 
tance between  the  points  be  small.  It  is  also  liable  to  error,  from 
a  want  of  accurate  previous  adjustment  in  the  instrument,  and 
from  the  bubble  not  being  exactly  in  the  middle  of  the  tube.  The 
latter  may  be  allowed  for,  by  noting  the  divisions  that  point  out 
the  position  of  the  level,  and  thence  calculating  the  effect  of  the 
inclination  of  the  axis  of  the  telescope. 

All  these  sources  of  error  may  be  in  a  great  degree  compen- 
sated by  the  second,  and  more  usual  method.  In  this,  the  instru- 
ment is  placed  as  nearly  as  possible  at  an  equal  distance  from  each 
of  the  points,  whose  difference  of  level  is  to  be  ascertained.  A 
staff  and  sliding  vane  is  set  up  at  each  of  these  points,  and  the  tele- 
scope directed  to  them  in  turn.  The  lines  upon  the  vanes  having 
been  made  to  coincide  with  the  horizontal  wire  of  the  telescope, 
the  difference  in  the  altitudes  of  these  lines,  above  the  points  on 
which  the  staves  rest,  is  the  difference  of  the  respective  levels. 
If  the  place  of  observation  is  exactly  equidistant  from  the 
two  points,  the  sphericity  of  the  earth  will  equally  affect  both 
vanes,  and  the  apparent  difference  will  be  the  same  as  the  true.  So 
also  if  the  axis  of  the  telescope  be  from  any  cause  inclined,  or  the 
line  of  collimation  do  not  coincide  with  it,  the  errors  that  arise 
will  be  equal  in  both  directions. 

When  a  line  of  considerable  length  is  to  be  levelled,  it  should 
be  divided  into  parts  by  stations  at  equal  distances;  from  180  to 
200  feet  is  a  convenient  space  for  this  purpose.  Staves  being  set 
up  at  the  two  first  stations,  the  levelling  instrument  is  placed  at  an 
equal  distance  from  each  of  them,  and  the  observations  made,  as 
has  been  directed  :  the  staff  from  the  first  station  is  then  moved 
forward  to  the  third  ;  that  at  the  second  remaining  stationary.  The 
instrument  is  next  moved  for  ward  to  a  point  equidistant  from  the 
second  and  third  station,  and  a  new  pair  of  observations  made.  In 
recording  the  observations,  the  heights  of  the  vanes,  as  seen  from 
each  position  of  the  instrument,  are  arranged  in  two  columns  ;  the 
heights  observed  by  looking  towards  the  first  station,  or  back- 
wards, being  set  in  one  column  ;  those  obtained  by  looking  for- 
wards in  the  other.  The  difference  between  the  sums  of  the  num- 
bers in  the  columns,  gives  the  difference  of  level  of  the  extreme 
stations. 


320  EQUILIBRIUM  OP  [Book  V. 

322.  The  mason's  level  is  composed  of  a  ruler,  to  which  a  plumb- 
line  is  adapted,  a  line  being  drawn  to  mark  the  proper  position  of 
the  cord,  and  an  opening  made  to  receive  the  plummet.  A  second 
ruler  is  adapted  to  the  lower  part  of  the  first,  whose  lower  edge 
is  at  right  angles  to  the  line  with  which  the  cord  of  the  plumb-line 
is  made  to  coincide  ;    when   the  latter  is  brought  to  its  proper 
position,  the  lower  edge  of  the  ruler  is  of  course  level,  being  at 
right  angles  to  the  plumb-line,  and  consequently  to  the  direction 
of  gravity. 

A  modification  of  this  instrument,  that  is  useful  in  many  cases, 
may  be  constructed  by  suspending  the  plumb-line  from  a  point  at 
the  intersection  of  two  bars  of  equal  lengths,  which,  therefore, 
form  sides  of  an  isosceles  triangle  ;  the  position  of  the  plumb- 
line  is  determined,  by  making  it  coincide  with  the  middle  of  a 
bar,  that  is  parallel  to  the  line  that  would  constitute  the  base  of 
the  triangle.  Levels  of  this  last  form  are  sometimes  constructed 
of  considerable  size,  the  bars  being  six  or  seven  feet  in  length. 
They  may  be  used  for  laying  out  ditches  and  trenches,  for  the 
distribution  of  water  for  agricultural  purposes,  but  are  not  suffi- 
ciently accurate  for  the  purposes  of  engineers. 

323.  The  correction  for  the  sphericity  of  the  earth,  applied  to 
the  first  method  of  observation  described  in  the  preceding  section, 
is  obtained  by  considering  the  horizontal  distance  as  a  mean  pro- 
portional between  the  earth's  diameter,  and  the  distance  of  the 
point  at  which  the  tangent  cuts  the  diameter  that  passes  through 
the  point,  whose  level  is  to  be  determined,  from  the  surface  of  the 
earth.     This  hypothesis  is  no  more  than  an  approximation,  but  is 
sufficiently  near  the  truth,  to  be  employed  on  most  occasions. 

Upon  this  hypothesis,  if  a  be  the  horizontal  distance  between 
the  two  points  ;  D  the  earth's  diameter ;  and  /i  the  height  of  the 
horizontal  vane  above  the  level  of  the  point  at  which  the  instru- 
ment is  placed,  all  estimated  in  the  same  unit  of  lineal  measure, 

*=£.  (355) 

324.  Observations,  made  with  a  levelling  instrument  at  one  of 
the   points,  whose  relative  heights  are  to  be  determined,  may 
also  be  affected  by  atmospheric  refraction.     This  is  not  the  case 
when  the  instrument  is  placed  half  way  between  them.  The  cor- 
rection to  be  used  in  the  former  case  is  not  properly  an  object  of 
investigation  in  mechanics,  as  it  is  derived  from  the  principles  of 
experimental  physics. 

325.  When  a  homogeneous  gravitating  liquid,  instead  of  being 
contained  in  a  single  vessel,  or  collected  in  a  great  mass  in  hol- 
low basins  on  the  surface  of  the  earth,  is  placed  in  bent  tubes,  or 


V.~\  GRAVITATING  LIQUIDS.  321 

in  vessels  communicating  with  each  other  at  bottom ;  the  sur- 
faces of  the  liquid  in  the  several  vessels  or  branches,  will  all  form 
portionsof  the  same  general  surface  oflevel ;  or  will, within  a  small 
space,  be  equally  elevated  or  depressed  below  the  same  horizon- 
tal plane.  For : 

If  in  the  first  place  we  suppose  the  liquid  to  be  contained  in  a 
tube  or  vessel,  with  no  more  than  two  branches,  we  may  consider 
it  as  divided  into  two  columns,  by  an  imaginary  surface,  where 
the  branches  communicate.  In  order  that  equilibrium  shall  exist, 
the  pressures  upon  this  surface,  exerted  by  the  two  columns,  must 
be  equal.  Now,  the  fluid  pressure  upon  a  surface  is  a  function  of 
the  vertical  co-ordinate  z,  §314,  or  will  depend  upon  the  vertical 
depth  of  the  surface  pressed,  beneath  the  horizontal  surface  of  the 
liquid.  This  quantity,  z,  must  therefore  be  equal,  in  the  case  of 
equilibrium,  in  both  branches,  and  the  respective  surfaces  must  be 
on  the  same  level.  The  same  mode  of  investigation  will  apply  to 
the  case  of  three,  or  any  number  of  branches  or  vessels,  commu- 
nicating beneath  the  surface  of  the  fluid. 

If  two  heterogeneous  fluids,  incapable  of  being  mixed,  be  placed 

in  the  two  branches  of  a  bent  tube,  the  pressure  exerted  by  one  of 

them  on  the  surface  of  contact,  will  be  by  the  integration  of  (352), 

p=fsdz>,  (356) 

and  if  s'  be  the  density  of  the  second  fluid,  and  z'  its  depth,  in  the 
case  of  equilibrium,  we  have  in  the  same  manner, 

p=fs'dzf  .  (357) 

In  the  case  of  liquidity,  when  each  of  these  fluids  may  be  con- 
sidered of  uniform  density  throughout,  we  obtain  by  the  integra- 
tion of  the  second  members  of  the  above  equations,  both  of 
which  are  equal  top  of  (346), 

AZ=*V ; 
and 

z  :  z'  :  :  s'  :  *  ;  (358) 

hence  : 

326.  The  heights  to  which  columns  of  heterogeneous  liquids 
rise  above  the  common  surface  of  contact,  are  inversely  propor- 
tioned to  their  respective  densities. 

If  two  immiscible  liquids  be  introduced  into  the  same  vase, 
the  denser  of  the  two  will  occupy  the  lower  part  of  the  vase ;  for 
the  freedom  of  motion  which  the  particles  of  each  possess,  will 
allow  it  to  descend,  by  its  superior  gravity,  through  the  rarer. 
Hence,  if  the  above  proposition  be  submitted  to  the  test  of  experi- 
ment, the  denser  liquid  must  be  first  introduced  into  the  tube  ; 
for  if  the  rarer  be  first  introduced,  the  denser  will  descend 
through  it,  separate  it  into  two  columns,  and  occupy  the  bend  of 
the  tube.  When  the  denser  liquid  has  been  introduced  into  the 
bent  tube,  the  rarer,  being  poured  into  one  of  the  branches,  will, 

41 


322  EQU1LIBK1UM  07  [Book    V* 

by  its  pressure,  force  down  the  level  of  the  denser  in  that  branch, 
while  that  in  the  other  will  be  elevated  ;  but  the  depression  will 
never  carry  the  surface  of  contact  beyond  the  bend  in  the  tube, 
which  will,  therefore,  continue  to  be  occupied  by  the  denser  fluid. 
The  surface  of  contact  must  be  level,  for  the  depths,  z,  andz', 
are  constant  quantities. 

327.  When  a  solid  body  is  wholly  immersed  beneath  the  sur- 
face of  a  fluid,  it  occupies  a  space  identical  with  what  it  did  be- 
fore immersion  ;  for  mere  immersion  does  not  alter  its  volume. 
The  whole  volume  of  the  fluid  mass,  when  it  contains  the  solid, 
will,  therefore,  be  increased  as  much  as  is  equal  to  the  volume  of 
the  solid. 

As  the  volumes  of  bodies  of  equal  weights  are  inversely  as  their 
densities,  the  weights  of  fluid,  displaced  by  bodies  of  different 
densities  wholly  immersed,  will  also  be  inversely  as  their  respec- 
tive densities. 

When  a  body  is  thus  immersed,  its  own  weight  tends  to  cause 
it  to  descend  in  the  direction  of  the  force  of  gravity  ;  but  this 
tendency  will  be  counteracted,  in  a  greater  or  less  degree,  by  the 
action  of  the  fluid.  If  instead  of  a  solid  body,  we  consider  the 
case  of  one  of  the  elementary  particles  of  the  fluid,  and  ascribe  to 
it,  upon  the  principle  of  §  313,  the  figure  there  assumed  for  one 
of  the  particles  of  the  fluid :  the  pressures  of  the  fluid  upon  the 
four  vertical  faces  of  the  parallelepiped,  will  exactly  counter- 
balance each  other ;  the  pressure  on  the  upper  surface,  will  be 
measured  by  a  column  of  the  fluid,  (346),  whose  area  is  that  of 
this  surface  of  the  particle,  and  whose  altitude,  is  the  vertical  depth 
of  the  same  surface,  beneath  the  level  of  the  fluid  (considered  as  ho- 
mogeneous); this  depth  in  the  case  of  a  liquid,  or  other  homoge- 
neous fluid,  will  be  the  actual  depth.  The  downward  pressure  on 
the  lower  base  will  be  the  sum  of  the  weight  of  the  superincum- 
bent column,  and  the  weight  of  the  solid  particle  itself,  while  the 
upward  pressure  on  the  same  base  will  be  measured  by  a  column 
of  the  fluid,  whose  altitude  is  the  vertical  depth  of  this  lower  sur- 
face. As  the  solid  particle  identically  replaces  in  bulk  an  equal  vol- 
ume of  the  liquid,  the  difference  of  these  two  pressures  will  be  the 
difference  of  the  weights  of  the  two  particles ;  this,  if  the  densities 
of  the  two  be  equal,  will  be  =0.  The  same  will  also  be  true,  as 
will  appear  from  the  same  section,  §  31 3,  whatever  be  the  direction 
of  the  surfaces  of  the  particle.  Now,  as  the  solid  body  is  made 
up  of  its  particles,  what  is  true  in  respect  to  one  of  them  in  this 
instance,  will  be  true  of  its  whole  mass,  and  the  tendency  of  the 
solid  body  to  descend,  under  the  action  of  gravity,  will  be  dimi- 
nished by  a  force,  whose  measure  is  the  weight  <5f  a  volume  of 
the  fluid,  equal  to  the  bulk  of  the  solid  body.  Hence,  if  the  solid 


Book   V.]  GRAVJTATIN*  LIQUIDS.  323 

have  the  same  density  as  the  fluid,  it  will  remain  at  rest  in  any 
part  of  the  latter  in  which  it  may  be  placed  :  if  its  density  be 
greater,  it  will  descend  ;  and  if  its  density  be  less  than  that  of 
the  liquid,  the  solid  will  rise  ;  but  the  force  with  which  it  moves, 
in  either  case,  will  have  for  its  measure  the  difference  in  the 
weights  of  equal  volumes  of  the  two  substances. 

It  will  therefore  be  seen,  that  when  a  solid  body  is  immersed 
in  a  fluid,  it  appears  to  lose  as  much  of  its  weight  as  is  equal  to 
the  weight  of  a  mass  of  fluid  whose  volume  is  the  same  with  its 
own.  This  loss  of  weight  was  by  some  considered  as  actual,  and 
not  merely  apparent.  This,  however,  is  not  the  case,  for  the 
tendency  to  descend  under  the  action  of  the  attraction  of  gravi- 
tation is  not  destroyed,  but  merely  opposed  ;  the  weight  of  the 
body,  which  is  the  sum  of  the  gravitating  forces  exerted  upon 
its  several  particles,  §  105,  still  remains ;  but  these  forces 
are  opposed  by  others  acting  in  a  contrary  direction,  and  their 
joint  resultant  is  of  course  less  than  the  weight  of  the  body.  To 
render  this  more  clear,  if  a  vase  be  taken  that  contains  a  liquid, 
and  if  a  solid  body  be  immersed  in  it;  although  the  latter  will 
appear  to  lose  a  portion  of  its  weight,  the  joint  weight  of  the 
vase,  the  liquid  and  the  solid,  will  still  be  the  same,  as  if  the  lat- 
ter were  weighed  separately,  and  its  weight  added  to  the  joint 
weight  of  the  other  two  substances. 

328.  When  the  solid  has  less  density  than  the  liquid,  the  re- 
sultant of  the  two  sets  of  forces  will  be  negative,  and  the  appa- 
rent loss  of  weight  of  the  solid  will  be  greater  than  its  own 
weight :  it  will  in  consequence  rise  to  the  surface  of  the  li- 
quid. Having  reached  the  surface,  it  will  continue  to  rise,  and 
elevate  a  portion  of  its  volume  above  the  surface,  until  equili- 
brium between  the  two  sets  of  forces  be  restored.  Thedownward 
pressure  on  the  lower  surface  becomes  in  this  case  no  more  than 
the  weight  of  the  solid  itself;  the  upward  pressure,  in  order  to 
be  equal  to  this,  must  have  for  its  measure  columns  of  fluid, 
whose  joint  weight  is  equal  to  that  of  the  solid  ;  and  the  joint 
volume  of  these  columns  will  therefore  be  equal  to  the  volume  of 
the  portion  of  the  solid  immersed.  Hence,  when  a  solid  body 
of  less  density  than  that  of  the  liquid,  has  risen  to,  or  is  placed 
upon,  the  surface  of  the  latter,  it  will  float  there,  displacing  as 
much  of  the  liquid  as  is  equal  in  weight  to  the  whole  weight  of 
the  solid  body. 

The  force  which  acts  downwards,  being  the  weight  of 
the  solid  body,  has  for  its  point  of  application  the  centre  of  gra- 
vity of  the  solid  ;  the  force  that  acts  upwards  is  the  resultant  of 
a  number  of  parallel  forces  exerted  by  an  infinite  number  of  ver- 


324  EQUILIBRIUM,  &c.  [Book  V. 

tical  columns  of  the  liquid,  which  together  make  up  a  volume 
equal  to  that  of  the  part  of  the  solid  that  is  immersed.  The  re- 
sultant of  such  a  system  of  forces  will  have  for  its  point  of  appli- 
cation the  centre  of  parallel  force  ;  this  being  determined  upon, 
the  principles  of  §  105,  will  be  the  same  with  the  centre  of  gra- 
vity of  the  part  of  the  solid  immersed.  And  as  it  is  necessary, 
not  only  that  two  opposing  forces  in  equilibrio  should  be  equal 
in  intensity,  but  that  they  should  act  in  the  same  straight  line, 
this  condition  is  also  necessary  for  the  the  existence  in  equili- 
brium ;  and, 

A  solid  will  float  in  equilibrio  upon  the  surface  of  a  liquid, 
when  it  has  displaced  a  volume  of  the  liquid  whose  weight  is 
equal  to  its  own,  and  when  the  centres  of  gravity  of  the  whole 
solid  and  of  the  part  immersed  are  situated  in  the  same  vertical 
line. 

329.  If  the  same  solid  body  be  placed  in  succession  in  two  dif- 
ferent liquids,  whose  densities  are  both  greater  than  its  own,  it 
will  float  at  the  surface  of  both.  The  part  of  the  solid  immersed 
will  displace  equal  weights  of  the  two  liquids,  but  the  volumes 
displaced  will  be  different,  if  their  densities  be  not  the  same. 
Now,  as  the  volumes  of  equal  weights  of  different  bodies  are  in- 
versely as  their  respective  densities,  the  parts  of  the  same  solid 
that  are  immersed  in  two  different  liquids,  on  whose  surface  it 
successively  floats,  are  inversely  as  their  respective  densities. 

If  the  floating  body  have  the  form  of  a  prism,  and  float,  in  con- 
formity with  the  second  of  the  above  conditions  of  equilibrium, 
with  its  axis  in  a  vertical  position  ;  as  the  horizontal  area  of  the 
part  immersed  will  be  constant,  the  body  will  be  immersed  to 
depths  measured  vertically  along  its  axis,  that  are  inversely  pro- 
portioned to  the  densities  of  the  liquid  on  which  it  floats. 


Book  V.}  PRESSURE,  &c.  325 


CHAPTER  III. 

y 

OF  THE  PRESSURE  OF  GRAVITATING  LIQUIDS. 

330.  The  pressure  of  gravitating  liquids  upon  surfaces  im- 
mersed  in  them,  or  upon  the  sides  of  the  vessels  that  contain 
them,  may  be  easily  determined  from  the  principles  of  the  pre- 
ceding chapter. 

The  equation  (352),  gives  us  for  the  value  of  p, 


and  if  the  origin  of  the  co-ordinate,  z,  be  at  the  surface  of  a  homo- 
geneous liquid, 

p—sz  ; 

substituting  this  value  in  the  expression  (346),  for  the  pressure, 
P,  we  have 

dP=dksz. 

The  measure  of  the  pressure  upon  an  infinitely  small  surface, 
therefore,  is  the  weight  of  a  column  of  the  liquid,  whose  base  is 
equal  to  the  surface  pressed,  and  altitude  equal  to  its  depth  be- 
neath the  surface  of  the  fluid. 

Upon  a  plane  surface  of  determinate  magnitude,  lying  in  a  ho- 
rizontal position,  the  whole  pressure  is  the  sum  of  the  partial  pres- 
sures upon  all  its  elementary  portions,  and  will  therefore  be 

P=te,  (359) 

•3  f*v  or  will  be  measured  by  the  weight  of  a  prism  of  the  liquid,  whose 
base  is  equal  to  the  surface  pressed,  and  whose  altitude  is  equal 
to  the  distance  of  the  plane  from  the  horizontal  surface  of  the  li- 
quid. If  the  plane  be  inclined  to  the  horizon,  call  the  distances 
of  its  respective  elements  from  the  surface  of  the  liquid,  z7,  z", 
z",  &c.,  the  sum  of  the  partial  pressures  will  be 


But  if  we  call  the  distance  of  the  centre  of  gravity  of  the  plane 
beneath  the  level  surface  of  the  liquid,  Z, 


and 

P=skZ  ;  (359a) 

and  this  will  be  true  of  any  surface,  whether  plane  or  curved. 

These  several  expressions,  applied  to  the  case  of  a  liquid,  con- 
tained in  a  vessel,  are  wholly  independent  of  the  bulk  of  the  li- 
quid itself;  for  they  include  no  other  quantities  than  the  area  of 
the  surface  pressed,  and  the  depth  of  the  liquid.  Hence  : 


TREISFRK  OP  [Book  V. 

33  J .  The  pressure  of  a  liquid,  upon  a  horizontal  plane,  is  equal 
to  the  weight  of  a  prism  of  the  liquid  whose  section  is  equal  to 
the  area  of  the  plane  ;  and  whose  altitude  is  equal  to  the  depth  of 
the  plane  beneath  the  surface  of  the  liquid. 

The  pressure  of  a  liquid  upon  a  surface  that  is  not  horizontal, 
is  equal  to  the  weight  of  a  prism  of  the  liquid,  whose  section  is 
equal  to  the  area  of  the  surface  pressed,  and  whose  altitude  is 
equal  to  the  depth  of  the  centre  of  gravity  of  that  surface  beneath 
the  level  surface  of  the  liquid. 

Upon  equal  and  similarly  situated  surfaces,  the  pressures  are  as 
the  depths  of  the  liquid  ;  and  at  equal  depths,  the  pressures  are 
as  the  surfaces. 

Upon  a  given  surface,  the  pressure  depends  wholly  upon  the 
depth  of  the  liquid  above  its  centre  of  gravity,  and  has  no  refe- 
rence to  the  quantity  of  liquid  ;  and  thus,  in  vessels  having  equal 
bases,  and  in  which  the  liquid  stands  at  equal  heights,  the  pres- 
sures on  the  bases  are  equal,  however  different  may  be  the  res- 
pective capacities  of  the  vessels. 

If  the  vessel  be  prismatic,  and  the  base  horizontal,  the  pressure 
on  the  base  is  exactly  equal  to  the  weight  of  the  liquid  it  contains  ; 
but  if  the  vessel  be  wider  at  top  than  at  bottom,  the  pressure  on  the 
base  is  less  than  the  weight  of  the  liquid  it  contains  ;  while  if  the 
vessel  be  narrower  at  top,  the  pressure  on  the  base  exceeds  the 
whole  weight  of  the  liquid. 

332.  When  a  liquid  is  in  equilibrio  in  a  vase,  its  surface  is,  as 
has  been  shown,  §  315,  horizontal  :  on  the  sides  of  the  vessel, 
equilibrium  exists  in  consequence  of  the  fluid  pressure  being  ex- 
actly counteracted  by  the  resistance  of  the  solid  boundary.  This 
resistance  may  be  represented  by  a  force,  whose  direction  is  a 
normal  to  the  surface  :  hence  the  liquid  pressure  must  also  act  in 
the  normal,  and  will  always  have  a  direction  perpendicular  to 
the  point  of  the  surface  on  which  it  acts.  In  addition  to  the  pres- 
sure on  the  bottom  of  the  vessel,  the  sides  are  also  pressed  by 
forces,  estimated  as  has  been  stated  above  ;  and  thus  in  a  pris- 
matic vessel,  not  only  will  the  base  sustain  a  pressure  equal  to 
the  whole  Weight  of  the  fluid,  but  the  sides  will  also  sustain  pres- 
sures perpendicular  to  their  several  surfaces :  the  measure  of  these 
forces  is  a  weight  of  a  prism  of  the  liquid,  whose  horizontal  section 
is  etfual  to  the  area  pressed,  and  whose  altitude  is  equal  to  the  depth 
at  which  the  centre  of  gravity  of  the  sides  lies  beneath  the  horizon- 
tal surface  of  the  liquid.  Thus,  in  a  cube,  filled  with  liquid,  the 
pressure  on  the  base  is  equal  to  the  whole  weight  of  the  liquid, 
and  acts  vertically  in  the  same  direction  with  the  force  of  gravity. 
On  each  of  the  four  vertical  faces,  the  pressure  is  equal  to 
half  the  weight  of  the  liquid  mass,  and  acts  in  a  horizontal  di- 


Book   V.~\  GRAVITATING  LIQUIDS.  327 

rection.  A  liquid,  therefore,  acting  on  the  sides  and  base  of  a 
vessel,  both  changes  the  intensity  and  the  direction  of  the  force 
which  acts,  namely,  the  weight  of  the  mass  itself  impelled  by  the 
force  of  gravity  in  a  vertical  direction. 

A  liquid  then,  is,  in  fact,  by  the  definition  of  §  131,  a  machine  ; 
and  masses  of  liquid  may,  and  frequently  are,  used  to  produce  ef- 
fects analogous  to  those  produced  by  machines. 

333.  In  a  close  vessel,  to  which  a  lateral  tube  is  adapted,  if  both 
be  filled  with  a  liquid,  the  pressure  on  the  sides,  the  top  and  the 
base,  will  depend  upon  the  areas  of  these  substances,  and  on  the 
height  of  the  liquid  in  the  lateral  tube.     This  will  be  true,  what- 
ever be  the  respective  dimensions  of  the  vessel  and  the  tube  adapt- 
ed to  it ;  and  thus  any  quantity  of  liquid,  however  small,  may  be 
made  to  counterbalance  any  other  quantity  however  great.    This 
principle  is  usually  styled  the  Hydrostatic  Paradox. 

When  a  liquid  is  contained  in  a  cjose  vessel,  and  a  pressure  is 
exerted  upon  it  by  means  of  a  piston  fitted  tightly  to  an  opening 
made  in  one  of  its  sides,  this  pressure  may  be  represented  by  the 
weight  of  a  mass  of  fluid  resting  upon  the  piston  as  a  base ;  the 
action  of  the  piston  will  therefore  produce  the  same  effect  upon 
the  liquid  mass  as  a  column  of  liquid,  whose  area  is  equal  to  that 
of  the  piston,  and  whose  altitude  is  such,  that  the  weight  of  the 
prism  of  liquid  that  has  this  area  and  altitude,  shall  be  equal  to 
the  pressure  upon  the  piston  ;  the  pressure  exerted  by  the  piston 
will  therefore  be  equally  felt  upon  all  the  sides  of  the  vessel,  and 
will  upon  an  area  equal  to  that  of  the  piston,  be  equal  to  the 
whole  pressure  the  latter  exerts. 

334.  These  principles  may  be  subjected  to  the  test  of  experi- 
ment, which  fully  confirms  them  ;  and  experiments  are  of  import- 
ance in  this  department  of  mechanics,  inasmuch  as  we  know  no- 
thing of  the  nature  of  the  particles  of  which  fluids  are  composed, 
nor  of  the  manner  of  their  action  upon  each  other;  the  hypothe- 
sis on  which  we  have  proceeded  in  the  mathematical  investiga- 
tion of  the  conditions  of  equilibrium,  and  of  the  measure  of  pres- 
sure, is  obviously  inaccurate,  in  omitting  the  viscidity,  and  the 
compressibility  of  liquids.     When,  however,  we  find  that  the 
deductions  from  the  hypothesis  are  exactly  consistent  with  ex- 
periment; we  infer  that  the  neglect  of  these  circumstances  pro- 
duces no  sensible  error.     The  experimental  illustrations  are  as 
follow : 

(1.)  If  water  or  any  other  liquid  be  contained  in  an  open  vessel, 
it  assumes  a  horizontal  surface,  except  so  far  as  it  is  influenced  near 
the  sides,  by  a  force  that  will  be  hereafter  the  object  of  investi- 
gation. 


328  PRESSURE  OF  [Book  V. 

(2.)  If  two  or  more  immiscible  liquids  be  poured  into  the  same 
vessel,  they  arrange  themselves  in  the  order  of  their  respective 
densities,  the  denser  liquids  occupying  the  lowest  places. 

(3.)  If  a  liquid  be  poured  into  a  bent  tube,  or  into  vessels  com- 
municating at  bottom,  it  rises  in  both  branches  of  the  tube,  and  in 
all  the  vessels,  to  a  common  level,  whatever  be  the  areas  of  the 
branches  of  the  tube,  or  the  cubic  contents  of  the  several  vessels. 

(4.)  Two  heterogeneous  liquids,  placed  in  opposite  branches  of 
a  bent  tube,  rise  to  heights  above  their  surface  of  contact  inverse- 
ly proportioned  to  their  respective  densities;  the  denser  liquid 
will  occupy  the  bend  of  the  tube,  and  the  surface  of  contact  will 
be  in  the  branch  occupied  by  the  rarer  liquid. 

(5.)  The  phenomena  of  bodies  denser  than  liquids,  in  which 
they  are  placed,  sinking;  and  those  which  are  rarer,  rising  and 
floating  at  the  surface,  are  too  familiar  to  need  illustration. 

(6.)  The  quantity  of  weight  lost  by  a  body  immersed  in  a  li- 
quid, may  be  shown  to  be  equal  to  the  weight  of  an  equal  volume 
of  the  liquid,  by  means  of  a  cylindric  bucket  which  a  solid  of  the 
same  form  exactly  fills.  If  the  bucket  and  solid  be  placed  in  one 
of  the  scales  of  a  balance  and  counterpoised,  and  the  solid  be  af- 
terwards suspended  from  the  scale,  in  a  vessel  containing  a  liquid, 
the  former  counterpoise  will  now  preponderate  :  but  if  the  bucket 
be  filled  with  a  portion  of  the  liquid, .equilibrium  is  restored. 

(7.)  The  quantity  of  liquid  displaced  by  a  floating  body,  may  be 
measured  by  performing  the  experiment  in  a  graduated  glass  jar  ; 
and  it  will  be  found,  that  the  increase  in  the  space  the  liquid  oc- 
cupies, is  exactly  equal  to  a  volume  of  the  liquid  whose  weight 
is  identical  with  that  of  the  solid  body. 

(8.)  The  same  experiment  performed  with  two  different  liquids, 
and  the  same  solid  body,  shows  that  the  quantities  the  latter  dis- 
places of  each  respectively,  are  inversely  as  their  densities. 

(9.)  Vases  of  different  figures  and  areas,  and  therefore  containing 
different  bulks  of  a  liquid,  may  be  adapted  by  screws  to  one  and 
the  same  base,  held  in  its  place  by  a  counterpoising  force  ;  and  it 
will  be  found  that  this  opposing  force  is  overcome  in  each  of 
them,  so  soon  as  the  liquid  acquires  the  same  depth.  The  same 
will  be  found  true,  whether  the  base  be  horizontal  or  inclined. 
The  force  that  keeps  the  base  in  its  place,  may  be  a  weight  act- 
ing through  the  intervention  of  a  lever  of  the  first  kind  ;  this 
weight  therefore  furnishes  a  measure  of  the  liquid's  constant  pres- 
sure in  the  vases  of  different  figures  ;  and  this  weight  will  be 
found,  if  the  lever  have  equal  arms,  to  be  equal  to  the  weight  of 
a  prism  of  water,  having  a  section  equal  to  the  base  on  which 
the  liquid  presses,  and  an  altitude  equal  to  the  depth  of  the  liquid 
in  the  vessel. 


•   •  f* 

Book   V.~\  GRAVITATING  LIQUIDS.  329 

(10.)  An  apparatus,  called  the  hydrostatic  bellows,  is  formed 
by  connecting  two  circular  or  elliptical  planks,  by  a  flexible 
band  of  leather,  in  such  a  manner  as  to  be  water  tight;  to  this 
a  tall  tube  is  adapted'.  If  a  liquid  be  poured  into  this  apparatus 
through  the  tube,  it  will  be  found  that  it  is  capable  of  counterpoi- 
sing a  weight  resting  upon  the  upper  plank,  equal  to  the  weight 
of  a  prismatic  mass  of  water,  whose  section  is  equal  to  the  area  of 
the  bellows,  and  altitude  equal  to  that  of  the  liquid  in  the  tube. 

(11.)  The  effect  of  pressures,  exerted  by  means  of  pistons,  may 
be  best  illustrated  by  the  machine  called  Bramah's  press,  or  by 
the  forcing  pump. 

335.  When  a  gravitating  liquid  is  placed  in  a  vessel,  the  pres- 
sure it  exerts  upon  equal  surfaces  of  its  sides,  will  increase  with 
the  depth,  as  is  obvious  from  the  investigations  of  §  314  ;  or  as 
may  be  easily  shown,  by  supposing  the  liquid  to  be  divided  into 
an  infinite  number  of  horizontal  strata. 

If  the  respective  distances  of  these,  from  the  surface,  be  z',  z", 
z'",  &c.,  the  respective  pressures  on  equal  surfaces,  will  be 

sz'dk,  sz"dk,  sz'"dk,  &c. 

quantities  continually  increasing  with  the  variable  co-ordinate,  z. 
Hence  : 

Although  the  pressure  upon  a  given  base,  horizontal,  inclined, 
or  vertical,  depends  upon  the  depth  to  which  its  centre  of  gravity 
is  immersed  below  the  level  of  the  liquid,  that  point  will  not  be, 
in  the  case  of  an  inclined  or  vertical  surface,  the  point  of  applica- 
tion of  the  resultant  of  the  fluid  pressures.  This  point  of  appli- 
cation is  called  the  Centre  of  Pressure,  and  must  be  lower  than 
the  centre  of  gravity  of  the  surface  pressed.  The  true  position 
of  the  centre  of  pressure,  may  be  thus  investigated  in  the  case 
of  a  plane  surface,  in  which  it  will  always  fall  in  the  plane  itself. 

Let  k  represent  the  area  of  the  surface,  dk  will  be  one  of  its 
small  elements  ;  let  z  represent  the  distance  of  the  centre  of  gra- 
vity of  the  whole  surface,  from  the  upper  level  of  the  liquid,  and 
z\  z",  z'",  &c.,  the  distances  of  its  several  elements. 

The  several  partial  pressures  will  be 

z  dk,  z'  dk,  z"  dk,  &c.  ; 
and  the  sum  of  their  moments  of  rotation,  §       , 


The  whole  pressure  will  be 

Zk;, 

and  if  we  call  the  distance  of  the  centre  of  pressure,  c,  the  mo*- 
ment  of  rotation  will  be 

Z&c; 

42 


330  PRESSUKE  OF  [Booh  V. 

and  this  will  be  equal  to  the  sum  of  the  moments  of  the  partial 
pressures,  or 

zkc=dk(.z2+z'-+z"a+&tc.)  ; 
whence 

-        :  (360) 


JLtK 

but  we  have 

Zk^dJ^z+z'+z"), 
and,  therefore, 


or  if  we  call  the  several  elements,  A,  B,  C,  &c.,  their  respective 
distances  from  the  surface,  o,  6,  c,  &c., 


A«+B6+Cc+&c.    ' 

a  formula  identical  with  (       ),  which  we  have  found  adapted  to 
?&e  investigation  of  the  position  of  the  centre  of  oscillation. 
This  may  be  given  in  a  more  convenient  form. 
Let  x  and  y,  be  the  vertical  and  horizontal  co-ordinates  of  the 
surface,  kthe-depth  of  its  upper  edge  beneath  the  level  of  the  li- 
quid, the  area  will  be 


the  several  distances,  z,  z',  z",  &c.  ; 

h+dx,  h+2dx,  h+3dx,  Sic.  : 
the  expression  (361)  will,  therefore,  become 


and  when  the  plane  that  is  pressed  by  the  liquid,  reaches  to  the 
level  surface  of  the  latter, 


(86* 
From  these  formulae  we  obtain  the  following  useful  results  : 

In  a  parallelogram,  whose  upper  side  is  on  a  level  with  the 
surface  of  the  liquid,  the  centre  of  pressure  is  at  a  distance  of  two 
thirds  of  its'height  beneath  that  surface. 

In  a  triangle,  whose  base  is  on  a  level'  with  the  same  surface, 
this  distance  is  one  half  its  altitude  ;  but  if  the  vertex  be  in  the 
surface  of  the  liquid,  the  distance  is  three-fourths. 

336.  When  a  solid  body  is  wholly  immersed  in  a  liquid,  its 
surfaces  undergo  pressures  ;  the  measures  of  these  will  be  the 
weight  of  columns  of  the  liquid,  having  the  surface  pressed  for  a 
base,  and  the  depth  of  its  centre  of  gravity  for  the  altitude.  The 
whole  amount  of  the  pressures  will,  therefore,  depend  upon  the 
magnitude  of  the  surface  of  the  solid,  and  have  no  reference  to  its 


Book 


GRAVITATING   LIQUIDS. 


volume  or  density.  But  when  the  directions  of  these  pressures 
are  taken  into  account,  they  have  a  resultant  which  depends  upon 
2  volume  solely,  and  is,  as  might  indeed  be  inferred  without 
investigation,  the  same  with  the  loss  of  weight  the  solid  experien- 
ces, or  is  equivalent  to  (he  weight  of  a  mass  of  the  fluid,  equal  in 
volume  to  the  immersed  solid. 

A  more  strict  investigation  is,  however,  necessary  : 
Let  the  body  immersed  be,  ABQ,  and  suppose  its  whole  mass 
to  be  divided  into  an  infinite  number  of  small  cylindrical  columns, 

lying  in  a  horizontal  position.  Let 
%  be  one  of  these,  terminated  at 
the  surface  in  the  bases,  B6,  Q^, 
and  distant  from  the  level  of  the  fluid 
by  the  depth  LB—z. 

Let  BN  be  a  normal  to  the  sur- 
face, B6,  which  may  be  considered 
as  plane.  Let  6H  be  the  projection 
of  this  surface  upon  a  plane  passing 
through  6,  and  perpendicular  to  BQ. 
6H  will  be  the  section  of  the  cylin- 
der, perpendicular  to  it3  axis,  and  6H~ -B6,  cos.  NBQ. 

The  liquid  exercises  upon  the  base,  B6,  a  pressure  whose 
measure,  taking  the  density  of  the  liquid=l,  is 

and  whose  direction  is  the  normal  BN.  If  this  pressure  be  de- 
composed into  three  rectangular  forces,  that  which  acts  horizon- 
tally in  the  direction  BQ,  will  be 

z.Bfc  cos.  NBQ=*./jH. 

In  the  same  manner  we  find  the  horizontal  pressure  on  the  oppo- 
site base,  Q</,  to  be 

*-#Q  ; 

but  the  surfaces,  6H  and  gQ,  are  equal  to  each  other,  and  the 
horizontal  pressures  are,  therefore,  equal  ;  they  are  also  opposite 
to  each  other,  and  hence  their  resultant  =0. 

And  in  like  manner  it  may  be  shown,  that  all  the  horizontal 
pressures  parallel  to  BQ,  mutually  destroy  each  other. 

If  we  next  suppose  the  body  to  be  divided  into  an  infinite  num- 
ber of  columns  also  horizontal,  but  at  right  angles  to  BQ,  the  same 
result  will  follow.  And  these  two  sets  of  forces  at  right  angles  to 
each  other,  make  up  the  whole  horizontal  pressures ;  for  each 
partial  pressure  may  be  resolved  into  two  forces  at  right  angles  to 
each  other,  and  all  of  these  are  included  in  the  two  sets  of  which 
we  have  spoken.  The  horizontal  pressures  upon  the  surface  of 
a  solid,  immersed  in  a  liquid,  are,  therefore,  in  equilibrio. 

Next  let  us  suppose  the  volume  of  the  solid  to  be  divided  into 
an  infinite  number  of  vertical  columns,  and  let  one  of  these  be 
A6,  terminated  by  the  bases,  Aa,  B6 ;  the  lower  base  being  im- 
mersed to  the  depth,  LB=z  ;  the  upper  to  the  depth  LA =z'. 


333  PRESSURE  OP,  &c.  [Book   V. 

The  normal  pressure  upon  the  upper  surface,  Aa,  resolved  into 
three  rectangular  forces,  gives  for  the  vertical  pressure, 

z'.AI. 
The  vertical  pressure  on  the  lower  surface  is  in  like  manner, 

z.VH, 
and 

AI=BH. 
The  resultant  of  these  two  opposite  vertical  pressures  is  therefor© 

0-*');BH, 

and  is  in  the  direction  BA.  It  is,  therefore,  equal  and  opposite 
to  the  weight  of  the  column  of  fluid  that  would  occupy  the  space 
of  the  column  A6.  And  the  resultant  of  all  the  vertical  pressures 
would  be  equal  and  opposite  to  the  weight  of  a  mass  of  the  liquid, 
equal  in  volume  to  the  solid  body  immersed. 

In  the  same  manner  it  may  be  shown  that  the  resultant  of  all 
the  pressures,  exerted  by  a  liquid  upon  the  vessel  that  contains 
it,  is  equal  to  the  weight  of  the  liquid  itself. 

This,  however,  does  not  contradict  the  law,  that  the  amount 
of  the  pressures  exerted  outwards  have  for  their  measure  the 
weight  of  a  column  of  the  liquid  whose  base  is  equal  to  the  whole 
surface  of  the  vessel,  and  whose  altitude  is  the  depth  of  its  centre 
of  gravity  beneath  the  surface. 


00k  V.}  SPECIFIC  GRAVITIES.  333 


CHAPTER  IV. 

OF  SPECIFIC  GRAVITIES. 

337.  The  specific  gravity  of  a  body  is  the  ratio  of  its  weight 
the  weight  of  an  equal  volume  of  some  other  body.     In  this 

jneral  sense,  it  is  equivalent  to  density,  which  is  the  relation 
jtween  the  weights  of  equal  volumes  of  different  bodies.  But 
hile  density  is  an  abstract  term,  and  is  determined  by  the  di- 
ict  comparison  of  the  bodies  in  question,  specific  gravity  is  re- 
tive,  and  is  a  numerical  value  of  the  density  obtained  by  com- 
irison  with  some  conventional  standard.  The  standard,  in 
;neral  use  for  this  purpose,  is  water.  As  this  substance  may 
mtain  gaseous,  earthy,  and  saline  impurities,  it  will  only  answer 
le  purposes  of  a  standard  when  freed  from  them.  This  may 
;  effected  by  distillation.  The  heat  of  boiling  drives  off  all  the 
iseous  matter,  and,  in  distillation,  the  solid  substances  are  left 
;hind  in  the  still.  Newly  fallen  rain  water,  at  a  distance  from 
*bitations,  or  that  obtained  by  the  melting  of  clean  snow,  is  also 
ifficiently  pure  for  the  purpose. 

338.  Water,  like  all  other  substances,  is  liable  to  changes  of 
jlume  by  changes  in  its  temperature  ;  hence,  it  can  only  be  em- 
loyed  as  a  standard,  at  some  conventional  temperature,  to  which 
le  results  of  the  experiments  for  determining  specific  gravities 
iust  be  reduced.     The  English  usually  take,  for  this  purpose,  a 
lean  atmospheric  temperature,  say  about  60°  ;  the  French,  the 
smperature  at  which  water  has  its  maximum  of  density.     The 
tter  is  by  far  the  most  convenient  and  scientific  method,  for  it 

hardly  possible  to  perform  experiments  at  the  exact  tempera- 
ire  chosen  by  the  English,  in  consequence  of  the  practical  diffi- 
alty  of  counteracting,  by  additions  of  colder  or  warmer  water, 
ic  constant  variations  in  the  temperature  of  the  air;  and  this 
icthod  is  dependent  for  its  accuracy  upon  the  indications  of  a 
lennometer,  the  divisions  upon  whose  scale  are  arbitrary.  On  the 
Lher  hand,  it  is  always  possible  to  obtain  water  at  its  maximum 
snsity,  and  easy  to  maintain  it  in  that  state.  Thus  a  vessel  of 
ater,  on  which  a  small  portion  of  ice  floats,  will  have  always  in 
s  lower  parts,  water  of  the  maximum  density  ;  for  so  long  as  this 
ate  has  not  been  attained  by  the  water,  the  cooler  portions  will 
nk  to  the  lower  part  of  the  vessel ,  in  consequence  of  their  greater 
eights  under  equal  bulks  ;  but  so  soon  as  the  maximum  of  den- 
ty  is  attained  by  the  lowest  portions  of  the  liquid,  this  descent 


•        SPECIFIC  ORATITIES.  [Book    T~. 

ceases.  Thus  the  French  standard  temperature  is  best  suited  to 
all  cases  where  great  accuracy  is  required,  as  it  is  better  to  obtain 
a  result  from  direct  experiments,  that  require  no  correction,  than 
to  correct  those  obtained  under  other  circumstances. 

In  the  cases  that  most  frequently  occur  in  practice,  such  nicety 
is  unnecessary,  and  the  experiment  may  be  performed  with  water 
of  any  temperature  ;  but  the  temperature  must  be  noted,  and  a 
correction  applied  for  it. 

This  correction  rests  upon  the  density  of  water,  at  the  experi- 
mental temperature.  We  therefore  subjoin  a  table  of  the  den- 
sities of  water,  at  different  sensible  heats,  the  maximum  of  den- 
sity being  employed  as  the  unit. 

TABLE. 

Of  the  Densities  of  Water  at  different  Temperature*. 


32°. 

0.9998918 

50°. 

0.9997825 

34°. 

0.9999428 

55°. 

0.9995324 

36°. 

0.9999761 

60°. 

0.99918S6 

38°. 

0.9999944 

65°. 

0.99S7549 

39°.4 

1.0000000 

70". 

0.9982239 

41°. 

0.9999950 

75. 

0.9976255 

43". 

0.9999739 

80°. 

0.9969316 

45°. 

0.9999378 

85°. 

0.9961497 

339.  The  densities  of  bodies  might  be  compared,  by  immersing 
them,  in  succession,  in  a  prismatic  vessel,  containing  a  liquid  of 
less  density  than  either.  The  liquid  in  which  a  solid  is  immersed, 
occupies  a  space  as  much  greater  than  it  did  before,  as  is  equal  to 
the  volume  of  the  solid.  Hence,  in  a  vessel  of  constant  and 
known  area,  the  relation  between  the  changes  of  level  caused  by 
the  immersion  of  different  solid  bodies,  gives  the  relation  between 
their  densities;  and  when  the  weight  of  a  given  bulk  of  water  is 
known,  the  same  method  would  give  their  respective  specific 
gravities  in  terms  of  water  as  the  unit. 

This  was  the  original  method  proposed  by  Archimedes,  in  the 
famous  problem  of  the  crown  of  Hiero. 

The  method  now  employed,  depends  upon  the  principle  that 
a  body,  when  immersed  in  a  fluid,  loses  as  much  of  its  weight  as 
is  equal  to  the  weight  of  an  equal  volume  of  the  fluid.  §  327. 

Let  S  be  the  specific  gravity  of  the  solid  body ;  *',  that  of  the 
liquid  ;  IP,  the  absolute  weight  of  the  solid ;  IP',  its  weight  in  the 
liquid  ;  ID — u?',  will  be  its  loss  of  weight  in  the  liquid,  and,  there- 
fore, the  weight  of  a  mass  of  the  liquid,  equal  in  volume  to  the 


Book  /^.]  SPECIFIC  GRAVITIES.  335 


solid.     If  B  be  this  common  volume,  the  definition  of  specific 
gravity  gives  us 

w  XM?  —  w' 

8=,  and  ''= 


whence 

w  w—w' 

*_*^*  f  (364) 

and 

8=,"       " 


rature, 


w — w'" 
If  the  liquid  employed  be  pure  water,  at  its  standard  tempe- 


and 

c» 


w — w' 


To  obtain  then  the  specific  gravity  of  a  solid  body,  its  weight 
must  be  divided  by  the  loss  of  weight  in  pure  water,  of  the  stand- 
ard temperature.  If  the  water  be  at  any  other  temperature,  we 
obtain  the  approximate  specific  gravity,  by  dividing  the  weight 
of  the  body  by  its  loss  of  weight  in  the  water;  this  may  be  re- 
duced to  the  true  specific  gravity,  as  will  be  seen  from  the  final  for- 
mula. (364),  by  multiplying  the  proximate  specific  gravity  by  the 
density  of  water,  at  the  temperature  at  which  the  experiment  is 
made  ;  the  density  of  water  at  the  standard  temperature  being 
taken  as  the  unit.  For  this  purpose,  the  table  in  the  preceding 
section  may  be  used. 

340.  In  determining  specific  gravities,  we  use  a  common  ba- 
lance, to  which  apparatus  intended  to  facilitate  the  process  of 
weighing  the  body  in  water,  is  adapted.  When  the  solids  are  in 
the  form  of  masses  united  by  the  attraction  of  aggregation  of 
their  particles,  and  are  denser  than  water,  the  process  is  ex- 
tremely simple.  The  body  is  weighed  in  air,  and  then  in  water, 
and  the  first  weight  is  to  be  divided  by  the  difference  of  the  two 
weights  ;  a  correction  is  then  applied^  as  has  been  just  stated,  for 
the  temperature  of  the  water,  if  it  be  not  at  its  maximum  density. 

When  the  solid  is  in  a  state  of  fine^powder,  it  must  be  placed  in 
a  vessel  whose  weight  in  air  and  in  water,  have  been  previously 
determined.  When  the  vessel  containing  the  powder  is  weighed 
in  air  and  in  water,  the  difference  between  these,  and  the  weights 
of  the  vessel  alone,  in  air  and  in  water,  are  the  weights  of  the 
powder  under  the  two  different  circumstances.  The  calculation 
may  then  be  performed  as  before. 


33C  SPECIFIC  GRAVITIES.  [Book    V. 

When  the  solid  is  soluble  in  water,  it  often  happens  that  a  li- 
quid may  be  found  in  which  it  is  not  soluble.  The  density  of 
this  liquid  being  known,  by  methods  hereafter  to  be  explained, 
the  body  may  be  weighed  first  in  air  and  then  in  this  liquid  ;  we 
obtain,  by  the  same  method  of  calculation,  its  specific  gravity,  in 
terms  of  the  liquid  as  the  unit.  This,  as  will  be  seen  from  (364) 
must  be  multiplied  by  the  specific  gravity  of  the  liquid,  in  order 
to  obtain  that  of  the  solid  in  terms  of  water  ;  for  it  is  obviously 
unimportant  whether  we  use  water  at  a  temperature  different  from 
the  standard,  or  a  liquid  whose  density  in  respect  to  water  is 
known. 

When  the  solid  is  less  dense  than  water,  its  weight  in  that 
liquid  cannot  be  obtained  directly  ;  but  it  may  be  attached  to 
another,  sufficiently  dense  to  cause  both  to  sink.  The  weight  of 
the  light  body  in  water  will  obviously  be  the  difference  in  the 
weights  of  the  two  bodies  when  united,  and  of  the  heavy  body 
alone. 

The  principle  and  method  of  calculation  ma}?  be  best  illustrated 
by  symbols  : 

Let  /  be  the  weight  of  the  light  body  ;  fe,  that  of  the  heavy  body, 
air  ;  h',  of  the  same  body  in  water  ;  c,  the  joint  weight  of  the  two 
bodies  in  air  ;  c',  their  joint  weight  in  water.  Then  c  —  c',  and 
h  —  h',  will  be  the  respective  losses  of  weight,  and  the  loss  of 
weight  of  the  light  body  will  be 

(o-O-(A-V); 
hence, 


The  divisory  in  this  formula,  will  always  exceed  the  dividend, 
and  the  resulting  specific  gravity  is  a  fraction,  which  is  usually 
obtained  in  the  decimal  form. 

When  no  convenient  fluid  can  be  found  in  which  the  solid 
is  insoluble,  its  specific  gravity  cannot  be  obtained  by  means  of 
the  hydrostatic  balance. 

341.  The  specific  gravity  of  a  liquid  may  be  determined  with 
the  hydrostatic  balance,  in  various  ways. 

(I.)  A  solid  of  convenient  size  and  form,  usually  a  bulb  of 
glass  of  known  weight,  may  be  weighed  in  water,  and  in  the 
liquid  whose  specific  gravity  is  sought.  The  respective  losses 
of  weight,  are  the  weights  of  equal  bulks  of  the  two  fluids;  and 
water  being  the  unit,  we  divide  the  loss  of  weight  in  the  liquid 
by  the  loss  of  weight  in  water. 

For  if  we  call  the  loss  of  weight  in  \vater,  w"  ;  the  loss  in  the 
other  liquid,  /,  tke  formula  (365)  becomes 

S=~  (367) 


!  '  ,-;p  *  || 

* 

, 


Book   f.]  SPECIFIC  GRAVITIES.  337 

(2.)  A  phial  of  known  weight  may  be  taken,  filled  with  water 
and  weighed  j  the  increase  of  weight  is  the  weight  of  the  con- 
tained water;  it  is  then  emptied,  and  filled  with  the  liquid, 
whose  weight  is  obtained  in  the  same  manner:  we  have  again 
weights  of  equal  bulks  of  the  substances,  whence  the  specific 
gravity  may  be  obtained,  as  in  the  preceding  instance. 

(3.)  We  know,  by  accurate  experiments,  the  weight  of  a  given 
bulk  of  water  ;  hence,  if  a  phial  of  known  weight  and  internal 
capacity  be  taken,  the  weight  of  the  water  it  contains  is  known. 
It  is,  therefore,  only  necessary  to  fill  the  phial  with  the  liquid 
whose  specific  gravity  is  sought,  and  weigh  it.  The  mode  of 
calculation  is  obvious. 

342.  In  determining  the  weight  of  a  given  volume  of  water,  or 
of  any  other  liquid,  it  is  found  much  more  easy  in  practice  to 
weigh  a  solid  of  known  dimensions  and  weight  in  the  liquid, 
than  to  make  a  vessel  of  a  given  capacity,  and  fill  it  with  the 
liquid.  This  arises  from  the  great  ease  and  certainty  with  which 
the  external  dimensions  of  a  solid  can  be  measured,  compared 
with  those  with  which  the  internal  capacity  of  a  vessel  can  be 
guaged. 

Sir  George  Shuckburgh,  in  order  to  ascertain  the  weight  of  a 
given  bulk  of  water,  (a  cubic  inch,)  in  Troy  grains,  made  use  of 
three  solids  of  the  same  material,  but  of  different  forms.  One 
was  a  sphere,  another  a  cylinder,  the  third  a  cube  :  the  material 
was  brass.  These  having  been  constructed  with'  every  possible 
care,  were  afterwards  measured  by  a  scale  furnished  with  pow- 
erful microscopes.  In  this  way  the  lineal  dimensions  were  as- 
certained, and  the  inequalities,  inseparable  from  the  best  mate- 
rials, and  most  accurate  workmanship,  detected  :  from  these  data, 
their  volumes  were  computed. 

Experiments,  with  the  same  three  solids,  were  also  made  by 
Kater,  in  the  researches  on  which  the  British  standard  of  weights 
and  measures  is  founded. 

In  the  French  investigations  for  establishing  the  basis  of  their 
metrical  system,  a  similar  method  was  used,  with  a  body  of  cylin- 
drical form,  and  of  larger  dimensions  than  those  used  by  Shuck- 
burgh. 

The  experiments  of  Kater  make  the  weight  of  a  cubic  inch  of 
pure  water  at  the  temperature  of  62°,  252  grs.  458,  Troy. 

The  State  of  New-York  has,  as  stated  in  Book  IV.,  Chapter 
V.,  taken  for  the  basis  of  its  standard  of  weight  the  avoirdupois 
pound,  of  such  magnitude  that  the  cubic  foot  of  pure  water,  at  its 
maximum  density,  weighs  1000  oz.  or  62£lbs. 

The  same  difficulty  that  we  have  stated,  exists  in  determining 
standards  of  cubic  measure,  by  means  of  the  internal  capacity  ; 

43 


333  srjECinc  GRAVITIES.  [Book  V. 

hence,  the  British  Government,  and  the  Legislature  of  the  State 
of  New-York,  have  defined  their  units  of  capacity,  by  prescribing 
the  number  of  pounds  of  distilled  water,  of  the  standard  tempe- 
rature that  the  vessel  shall  contain.  This  method  is  preferable 
to  that  of  defining  it  by  the  number  of  cubes  of  the  measure  of 
length  that  it  shall  comprise. 

343-  In  the  instructions  for  determining  specific  gravities  of 
§  33S,  it  will  have  been  noted  that  the  weight  in  air  is  spoken  of, 
instead  of  the  absolute  weight  referred  to  in  the  theory  of  §  337. 
These  are  not  identical,  for  the  air  being  a  fluid,  presses  upon  the 
solid,  and  produces  a  loss  of  weight  analogous  to  that  caused  by 
liquids,  as  demonstrated  in  §  325.  Indeed,  as  the  air  has  an  uni- 
form density  within  the  space  the  solid  under  experiment  occu- 
pies, the  effect  is  in  fact  identical,  and  the  solid  loses  as  much  of 
its  absolute  weight  as  is  equal  to  the  weight  of  its  volume  of  air. 
In  different  bodies,  of  equal  weight,  this  loss  will  be  in  the  in- 
verse ratio  of  their  densities,  and  thus  the  rarer  bodies  will  be 
the  most  affected.  In  very  accurate  investigations,  it  becomes 
necessary  to  take  the  buoyancy  of  the  air  into  account.  The 
principle  on  which  this  correction  is  founded,  may  be  inferred 
from  the  theory  that  has  already  been  laid  down.  The  exact 
amount,  and  the  variations  to  which  it  is  subject,  must  be  defer- 
red until  we  have  investigated  the  properties  of  air. 

When  the  weights  and  the  substance  to  be  weighed  are  homo- 
geneous, both  are  equally  affected  by  the  buoyancy  of  the  air. 
Hence,  there  area  few  cases  in  which  no  correction  is  necessary, 
even  in  the  most  accurate  investigations  :  thus,  in  prescribing  the 
weight  of  a  given  measure  of  distilled  water,  for  the  purpose  of 
defining  the  unit  of  weight,  the  British  and  New-York  statutes 
declare  that  the  experiments  shall  be  made  with  brass  weights, 
under  which  denomination  the  experimental  solid  is  included. 

344.  In  many  practical  cases,  methods  more  speedy  than  are 
furnished  by  the  hydrostatic  balance,  are  necessary.  This  neces- 
sity occurs  more  frequently  in  the  determination  of  the  specific 
gravities  of  liquids,  and  particularly  in  ascertaining  the  value  of 
spirituous  liquors,  both  for  the  convenience  of  commerce,  and  the 
collection  of  revenue.  In  these  cases,  an  instrument  called  the 
Hydrometer,  is  employed. 

A  hydrometer  consists  essentially  of  three  parts  :  a  stem,  on 
which  divisions  are  drawn;  a  bulb  or  hollow  float,  to  render  it 
buoyant  in  a  liquid  ;  and  a  weight  by  which  the  stem  may  be 
made  to  float  in  a  vertical  position.  These  parts  are  exhibited  in 
the  following  figure,  in  which  A  is  the  stem,  B  the  bulb,  and  C 
the  weight 


Book 


»PBCIFIC  GRAVITIES. 


339 


The  principle  on  which  the  hydrometer  is 
used,  is,  that  a  body  net  to  float  on  two  different 
fluids,  of  te.«s  density  than  itself,  will  displace 
volumes  of  them,  that  ore  inveiveJy  as  their  re- 
spective densities.  §330. 

It  is,  therefore,  obvious,  that  the  hydrometer 
must  he  less  dense  than  the  rarest  liquid  in 
which  it  is  intended  to  be  used  ;  and  that  in  the 
most  dense,  it  must  sink  at  least  so  far  as  to 
cover  its  bulb;  otherwise,  it  would  in  the  one 
case  sink  to  the  bottom,  and  in  the  other,  the 
liquid  would  not  reach  the  divisions  of  ihe  stem. 
Delicacy  of  indication  will  be  promoted  in 
the  hydrometer  by  making  the  stem  slender, 
in  which  case  the  differences  of  the  depths  to 
which  it  sinks  will  be  considerable  for  small 
changes  of  specific  gravity.  On  the  other  hand, 
the  extent  of  differing  specific  gravities,  or  what 
is  styled  the  scale,  will  be  limited,  when  the 
stem  is  slender,  unless  it  be  made  of  inconve- 
nient length.  When  great  accuracy  is  required, 
the  stem  must  be  necessarily  slender;  and  in 
order  to  obtain  measures  of  specific  gravities, 
not  included  within  the  scale  of  a  single  instrument,  two  or  more 
hydrometers  must  be  used. 

This  is  generally  the  case  with  the  hydrometers  used  by  che- 
mists, of  which  the  material  is  glass.  Glass  is  well  adapted  to 
the  purpose,  from  its  cleanliness;  the  case  with  it  may  be  fa- 
shioned by  the  blowpipe;  and  from  its  being  acted  upon  by  but 
few  chemical  liquids.  On  the  other  hand,  its  fragility  is  an  ob- 
jection to  its  use  by  manufacturers,  and  in  the  hands  oif  fiscalofii- 
cers. 

The  necessity  of  having  more  than  a  single  instrument,  may 
be  obviated  by  adapting  moveable  weights  to  the  same  hydro- 
meter, in  such  manner  that  they  may  be  removed,  or  added,  ac- 
cording to  the  density  of  the  liquid. 

For  these  reasons  the  hydrometers  in  more  frequent  use,  are 
made  of  metallic  substances,  and  have  metallic  weights. 

Dicas'  hydrometer,  which  is  prescribed  by  law  in  the  customs 
of  the  United  States,  has  36  moveable  weights,  and  the  stem  has 
10  divisions:  the  weights  are  so  adjusted  that  in  a  given  liquid, 
if  the  hydrometer  sink  with  one  of  the  weights  to  the  0  on  the 
stem,  the  next  weight  shall  sink  it  10°.  Hence  the  divisions 
amount  to  360.  This  instrument  is  extremely  delicate,  and  well 


340  SPECIFIC  CHAYITIES.  [Book    V. 

suited  for  nice  investigations,  but  is  too  complex  and  trouble- 
some in  its  use  for  ordinary  purposes. 

A  hydrometer  upon  similar  principles,  but  of  greater  simpli- 
city, has  also  been  constructed  in  Boston. 

A  still  more  simple  instrument  has  been  introduced  by  South- 
worth,  of  New  York.  The  material  is  also  silver,  and  there  is 
but  one  moveable  weight.  All  these  instruments  measure  spe- 
cific gravities  from  that  of  pure  water,  or  1,  to  that  of  alcohol  or 
0.825. 

Silver  is  objectionable  as  a  material  for  hydrometers,  unless  it 
be  gilt,  in  consequence  of  its  liability  to  tarnish,  and  the  necessity 
of  cleaning  it,  which  will  wear  away  the  metal,  and  alter  the  in- 
dications of  the  instrument. 

345.  The  hydrometer  may  be  used  for  determiningthe  weights 
and  specific  gravities  of  solids.  For  this  purpose  the  instrument 
is  made  so  much  lighter  than  water,  as  to  require  the  addition  of 
a  considerable  weight  to  sink  it  to  a  mark  on  the  middle  of  its 
stem.  This  weight  being  known,  in  some  conventional  unit  or 
standard,  the  body  to  be  weighed  is  placed  upon  a  cup  prepared 
for  the  purpose  at  the  top  of  the  instrument;  weights 
are  added  until  the  hydrometer  sink  to  the  mark  at 
which  it  before  stood  ;  the  joint  weight  of  the  body, 
and  the  additional  weights,  is,  therefore,  the  same  as 
that  which  was  found  by  experiment  to  bring  the  in- 
strument to  this  position.  It  only  remains  to  subtract 
the  weight  added  when  the  body  is  placed  in  the  cup, 
from  that  which  when  used  alone  sinks  it  to  the  proper 
level ;  the  difference  is  the  weight  required. 

If  the  specific  gravity  is  to  be  ascertained,  the  bo- 
dy is  next  placed  in  a  second  cup,  adapted  for  the  pur- 
pose, to  the  top  of  the  weight  that  steadies  the  instru- 
ment, where  it  is  of  course  immersed  in  the  water. 
Its  loss  of  weight  will  be  apparent  by  a  rise  in  the 
stem  of  the  instrument ;  fresh  weights  are  added  to 
bring  it  again  down  to  the  original  mark,  and  these  are 
the  measure  of  this  loss. 

Such  is  the  hydrometer  of  Nicholson,  the  form  and  arrange- 
ment of  which  may  be  understood  from  the  inspection  of  the  an- 
nexed figure. 

An  instrument  for  weighing,  founded  upon  similar  principles, 
but  far  superior  in  accuracy,  has  recently  been  constructed  by 
Mr.  Hassler,  to  be  employed  in  the  rectification  of  the  weights 
and  measures  used  in  the  customs  of  the  United  States.  A  hoi- 


BoOOk   V.}  SPECIFIC  GRAVITIES.  341 

low  bulb  of  glass,  of  the  figure  of  a  flask,  is  placed  in  a  vase  that 
is  set  upon  a  bracket ;  three  steel  rods  are  sealed  to  the  neck  of 
the  flask,  and  bear  a  metallic  plate,  cut  in  such  a  manner  as  to 
form  three  horizontal  arms  that  project  beyond  the  sides  of  the 
vase;  from  these  arms  three  rods  proceed  downwards,  two  of 
which  embrace  the  bracket,  and  the  third  descends  in  front  of 
it ;  these  rods  bear  a  shelf  or  scale  on  which  weights  and  the 
substance  to  be  weighed  are  placed.  By  the  use  of  the  bracket, 
the  position  of  the  weight  is  thus  brought  beneath  the  vase, 
and  the  equilibrium  of  the  apparatus  is  rendered  stable,  while  in 
Nicholson's  hydrometer  the  equilibrium  is  tottering.  The  opera- 
tion of  weighing  may,  therefore,  be  performed  with  greater  ease. 
The  size  of  the  instrument  is  also  much  greater  than  is  ever  given 
to  Nicholson's  ;  and  as  the  whole  exposed  to  the  liquid  is  of  glass, 
except  the  rods,  mercury  may  be  used  instead  of  water,  by  which 
liquid  the  capacity  of  the  instrument  to  bear  heavy  weights,  is  in- 
creased more  than  thirteen  fold. 

346.  The  specific  gravity  of  bodies,  is  one  of  their  most  im- 
portant distinctive  characters,  and  must  hence  be  included  among 
the  properties  by  which  they  are  described  and  defined.  By 
means  of  it,  we  may  frequently  determine  the  nature  of  bodies, 
without  having  recourse  to  any  other  means ;  and  in  all  cases  it 
is  an  aid  in  the  discovery  of  the  class  to  which  substances  hitherto 
unexamined  are  to  be  referred.  We  therefore  use- the  method  of 
specific  gravities  in  many  branches  of  physical  science,  in  several 
of  the  arts  ;  and  although  in  commerce  it  is  as  yet  applied  to  but 
few  purposes,  its  use  is  capable  of  much  farther  extension. 

The  seven  ancient  metals,  which  are  still  in  most  frequent  use, 
have  such  marked  differences  in  density,  that  they  may  frequently 
be  distinguished  from  one  another  by  their  specific  gravities  ;  and 
if  alloyed  or  adulterated,  the  determination  of  the  specific  gravi- 
ties of  the  compounds  will,  in  many  cases,  detect  the  mixture. 

Thus  in  gold,  whether  in  the  shape  of  bullion,  or  of  coin,  great 
specific  gravity  is  often  a  test  of  its  value.  Until  recently  it  was 
a  sure  one ;  for  gold,  before  the  discovery  of  platinum,  was  the 
densest  of  all  known  substances,  and  any  admixture  is  sure  to  pro- 
duce a  diminution  in  the  specific  gravity.  Recently  an  alloy  of 
platinum  has  been  discovered,  that  possesses  many  of  the  external 
characteristics  of  gold.  Platinum  being  the  densest  of  substances, 
we  now  know  that  even  the  appropriate  density  of  alloys  of  gold 
may  be  given  by  it,  to  the  alloys  of  which  the  former  metal  forms 
a  part.  These,  however,  want  the  ductility  and  malleability  of 
gold,  and  when  they  have  the  same  colour,  have  less  density  than 
its  alloys.  Among  the  other  ancient  metals,  tin  is  the  least  dense, 
and  hence  when  impure,  its  specific  gravity  is  increased. 


342  SPECIFIC   GRAVITIES.  [000k    V. 

The  adulterations  of  drugs,  and  pharmaceutical  preparations, 
may  be  frequently  detected  by  the  change  of  specific  gravity  they 
produce.  Acids,  in  particular,  can  have  their  puriiy  tested  in  al- 
most all  cases. 

In  the  operations  of  nractical  chemistry,  the  strength  of  solu- 
tions, the  proper  state  of  concentration  suited  for  crystallization, 
for  fermentation,  and  JistiJJation,  are  known  by  the  use  of  the 
hydrometer.  This  instrument  is  therefore  of  value  in  the  aits  of 
brewing,  distilling,  sugar  refining,  as  well  as  in  the  processes  that 
are  strictly  chemical.  •.-••*• 

The  value  and  character  of  metallic  ores,  may  also  be  in  many 
cases  inferred  from  their  specific  gravity  ;  and  in  general  it  forms  a 
marked  distinction  of  minerals,  by  means  of  which  they  may  be 
classed,  and  their  constituent  parts  inferred. 

Such  are  a  few  of  the  more  important  purposes  to  which  the 
method  of  specific  gravities,  is  applicable. 

347.  When  two  substances,  whose  specific  gravitiesare  known, 
ore  mixed  mechanically,  it  might  at  first  sight  be  inferred  that  the 
proportions  in  which  they  exist  in  the  compound,  would  be  rea- 
dily inferred  from  its  specific  gravity.  This  will  be  apparent 
from  the  following  investigation  : 

Let  w  and  iv'be  the  weights  of  the  two  bodies  ; 
#  and  g'  their  respective  specific  gravities  ; 
G  the  specific  gravity  of  the  compound. 

The  weigh*  of  the  compound  is  w+iv'.    The  weight  of  its  vol- 
ume of  water  (364)  is 

w+w' 

~~G™ 

•         The  respective  weights  in  water,  of  the  volumes  of  the  two  com- 
ponents, are 

w        .  to' 

7and7! 

hence 


=-4-—;  (368) 

*»        S    S 
whence  we  obtain 

w=± ^      (ic+i*Q  ;  369) 

and 

,_(£r— -G)^,     ,     ,,  (370^ 

*«  "^~  _     j f  u  -+"  x(,  )  *  v          / 

These  formula?  are,  however,  of  little  or  no  value  in  practice  ; 


SPECIFIC 


343 


for  when  two  bodies  are  mixed,  their  joint  volume  is  rarely  or 
never  the  same  as  the  sum  of  their  respective  volumes.  In  gene- 
ral, the  joint  volume  is  less  than  the  sum  of  the  separate  volumes. 
This  diminution  of  volume,  that  occurs  in  most  cases  of  mixture, 
is  called  Concentration.  The  most  remarkable  instance  of  con- 
centration that  has  been  noted,  is  that  stated  by  professor  Robin- 
son, who  says,  that  when  40  pts.  of  platinum  are  alloyed  with  5 
pts.  of  iron,  the  bulk  of  the  mass  is  but  39.  This  alloy,  then,  of 
the  densest  simple  substance  with  one  of  little  more  than  a  third 
of  its  density,  gives  a  compound  even  more  dense  than  the  first.  In 
the  case  of  the  mixtures  of  alcohol  and  water,  that  form  the  spi- 
rituous liquors  of  commerce,  the  concentration  produces  marked 
effects,  and  must  be  taken  into  account  in  all  the  estimates  of  their 
value  that  are  drawn  from  their  specific  gravities.  The  densities 
of  such  mixtures  have  been  made  the  subject  of  accurate  experi- 
ments by  Gilpin,  the  results  of  a  part  of  which  are  comprised  in 
the  following 

TABLE 

Of  the  densities  of  mixtures  of  alcohol  and  water,  at  the  temperature  of 
600  of  Fahrenheit. 


PTS.  OF 

WATER. 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
0 


PTS.  OF 

ALCOHOL. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 
10 
10 
10 
10 
10 
10 
10 
10 
10 
10 
10 


SPECIFIC  GRAVITIES. 

1.00000 

0.98654 

0.97771 

0.97074 

0.96437  * 

0.95804 

0.95181 

0.94579 

0.94018 

0.93493 

0.93002 

0.92449 

0.91933 

0.91287 

0.90549 

0.89707 

0.88720 

0.87569 

0.88208 

0.8456S 

0.82500 


344  SPECIFIC  GRAVITIES.  [Book   V. 

We  have  not  reduced  this  table  to  the  standard  of  the  maximum 
density  of  water;  for  the  habitual  custom,  not  only  in  England,  but 
in  France,  where  the  spirits  of  commerce  are  concerned,  is  to  re- 
duce the  densities  at  the  observed  temperatures,  to  that  of  60°. 
Upon  this  principle  the  hydrometer  or  alcoometer,  planned  by 
Gay  Lussac,  and  used  in  the  French  excise,  is  graduated. 

In  ascertaining  the  values  of  spirituous  liquors,  for  the  purpose 
of  the  collection  of  duties  in  the  United  States,  four  different 
stages  of  proof  are  established  by  law ;  each  of  these  pays  a  specific 
duty.  The  spirit  that  lies  between  any  two  of  these  in  density, 
is  counted  as  belonging  to  the  lower  proof.  This  method  is  un- 
fair in  practice,  and  has  been  made  the  source  of  actual,  if  not  of 
technical  frauds  upon  the  revenue.  The  true  and  equitable 
principle  is  that  adopted  by  the  French  government ;  by  this 
the  duty  is  estimated  upon  the  quantity  of  alcohol  of  the  specific 
gravity  of  0.825,  at  the  temperature  of  60°,  that  is  contained  in 
the  mixture. 

348.  One  other  method  of  determining  the  specific  gravities 
of  liquors  remains  to  be  mentioned.     A  number  of  bubbles  may 
be  blown  from  a  glass  tube,  having  different  densities.     Their 
specific  gravities,  having  been  determined  by  experiment,  in  li- 
quids of  known  densities,  are  marked  upon  them. 

When  the  specific  gravity  of  a  liquid  is  to  be  determined,  they 
are  thrown  into  it,  until  one  is  found  that  remains  quiescent  in 
any  part  of  the  liquid  in  which  it  is  placed,  and  with  the  specific 
gravity  of  this  bubble,  that  of  the  liquid  is  identical. 

349.  We  subjoin  a  table  of  specific  gravities  compiled  from  the 
most  accurate  authorities. 

TABLE 

Of  the  specific  gravities  of  bodies,  in  terms  of  Mater,  at  Us  maximum  of 

density. 

f  Rolled,         ....          22.6667 

I  Wire,  ....         21.0396 

Platinum,  <>  Hammered,  .         .         20.3356 

I  Purified,       .         .         .  19.4981 

i  Hammered,  .         .         .          19.3600 

Pure  gold.  J  Cast<  ....          19.2562 

Gold  22  carats  fine  cast,                      .         .  17.4846 

do    20,      do       "    cast,                    .  15.7075 

Tungsten,       ...                  .  17.6000 

Mercury, 13.5967 

Lead,               11.3512 

Palladium 11.3000 

Rhodium,                  1 


r.j  SPECIFIC  aRAVITIES. 

Silver  cast,      .         .      .->.  (,,•;»/        .         .  10.4743 

Bismuth,         .         .      -  {r-:j..v*         •         •  9.8220 

Copper,                 '*,.;;•*•>  =  ;"  i*-.       •         •  8.7880 

Molybdenum,        i.^    ..-^  '',-t.\..      .         , .;  8.6110 

Arsenic,           .         .      ,.  -.,       -.  f         .         •  8.3080 

Nickel,            .         .         .       ' .         .         .  8.2790 

Uranium, 8.1000 

Soft  Steel,    |  Hammered, 

(Cast,          .         .         .     ..:-;•.  r-  7.8324 
Hard  Steel,  I  Hammered, 

(  Cast,          ....  7.8156 

Cobalt,            .                   ....  7.8119 

Bar  Iron, 7.7873 

Tin, 7.2907 

Cast  Iron,        .                  *   -\ '.,-.         .         .  7.2083 

Zinc,      ....     ......       "»/.  6.8604 

Antimony,       ......  6.7120 

Tellurium, 6.1150 

Chromium,      ......  5.9000 

Iodine,             .     .  -.  J     ....  4.9480 

Sulphate  of  Baryta,        . '^;V -,'•'.',     .         .  4.4300 

Zircon,            ....     *'V  '     .  4.4157 

Ruby  and  Sapphire,  (oriental)    .         .     v  V  4.2830 

Diamond,  from         .....  3.5307 

to              3.5007 

Flint  Glass,              '  '  3.3290 

Fluor  Spar,              3.1908 

Tourmaline, 3.1552 

Native  Sulphur, 2.0329 

Sodium,          .         .                            .  0.9726 

Potassium,      ...                   .  0.8651 

Sulphuric  Acid,        .         .         .         .         .  1,8408 

Nitric  Acid, ,  1.5500 

Water  from  the  Dead  Sea,         .         .         •  1.2402 

Nitric  Acid,             1.2176 

Sea  Water, 1.2062 

Milk, 1.0300 

Distilled  Water,       .         .  1.0000 

Bourdeaux  Wine,     ...  0.9938 

Burgundy  Wine,      ...                  •  0.9214 

Olive  Oil,        .         .         .         .                  .  0.9152 

Muriatic  Ether,        .      "...     •         •  0.8740 

Spirits  of  Turpentine,        .         .         •  0.8697 

Naptha,          .         .         •     '   •         •         •  0.8475 
Standard  Alcohol  of  Gilpin,       . 
Alcohol  perfectly  pure, 
Sulphuric  Ether,      . 

44 


345 


346  SPECIFIC  GRAVITIES.  [Book  V. 

350.  If  the  weight  of  a  given  bulk  of  water  be  known,  the 
Hydrostatic  balance  may  be  applied  when  geometric  methods  fail, 
to  determine  the  volume  or  cubic  contents  of  bodies.  For  this  pur- 
pose they  must  be  weighed  in  air,  and  in  water,  and  as  their 
loss  of  weight  is  the  weight  of  their  volume  of  water,  their  cubic 
contents  may  be  calculated. 

Their  volume  B  in  cubic  inches,  when  the  weights  are  taken 
in  troy  grs.,  will  be 

w  —  w' 


Their  volume  in  cubic  feet,  when  the  weights  are  taken  in  avoirdu- 
pois ounces  will  be 


When  the  specific  gravity  of  a  body  is  known,  we  may  calcu- 
late its  volume  when  its  weight  is  given,  or  its  weight  when  its 
volume  is  given  upon  similar  principles. 

If  B  be  the  volume,  and  W  the  weight  of  the  body,  g  its  speci- 
fic gravity,  w  the  weight  of  the  cubic  unit  of  water  in  the  unit  of 
measure,  we  have 

W=B£W>,  (373) 

and 

B=—  .  (374) 

gto 


Book  V.]  ELASTIC  FLUIDS.  347 


CHAPTER  V. 

Oj-  THE  NATURE  AND  CHARACTERS  or  ELASTIC  FLUIDS,  AND  OF  THE 
PRESSURE  OF  THE  ATMOSPHERE. 

351.  We  become  acquainted  with  the  material  existence  of 
the  air  that  composes  the  atmosphere  of  our  earth,  and  which  is 
not  at  first  as  obvious  to  our  senses  as  that  of  solids  and  liquids, 
in  various  ways : 

(1.)  We  observe  it  moving  in  currents  of  wind,  capable  of 
producing  powerful  mechanical  effects  ; 

(2.)  We  find  it  resisting  the  entrance  of  other  substances  into 
the  space  it  occupies,  thus  :  when  a  glass  jar  is  inverted  and  press- 
ed into  a  mass  of  water,  the  water  enters  it  at  first  to  but  a  small 
distance ;  and  whatever  be  the  depth  to  which  it  is  forced  down, 
the  air  will  still  occupy  a  part  of  the  jar ; 

(3.)  We  may  weigh  it,  and  thus  show  that,  like  other  mate- 
rial substances,  it  is  affected  by  the  attraction  of  gravitation. 

The  fluid  nature  of  air  may  be  shown  by  the  freedom  with 
which  bodies  at  the  surface  of  the  earth  move  in  it ;  and  by  its 
exerting  pressures  in  all  directions,  and  therefore  upwards  as 
well  as  downwards  :  thus,  if  a  glass  vessel  be  filled  with  water, 
and  a  piece  of  paper  be  laid  upon  the  surface  of  the  water,  pro- 
jecting over  the  edges  of  the  vessel ;  the  vessel  may  be  carefully 
inverted  without  spilling  the  water ;  and  the  liquid  will  after- 
wards remain  in  the  vessel,  in  which  it  is  supported  by  the  pres- 
sure of  the  air. 

The  experiment  (2)  may  be  extended  to  show  another  es- 
sential property  of  air,  namely,  its  elasticity ;  for,  as  the  ves- 
sel descends  in  the  mass  of  water,  the  water  rises,  although,  as 
has  been  stated,  it  never  wholly  fills  it.  The  space  occupied  by 
the  air,  is  therefore  lessened  by  the  pressure  of  the  liquid  ;  if  the 
vessel  be  permitted  to  rise,  the  water  descends,  until  the  mouth 
of  the  vessel  reach  the  surface,  when  the  air  again  occupies  the 
same  space  it  did  at  first.  It  is,  therefore,  capable  of  having  its 
bulk  diminished  by  pressure,  and  of  restoring  itself  to  its  origi- 
nal volume  when  the  pressure  is  removed,  or  is  elastic. 

352.  This  elasticity  is  considered  as  permanent,  for,  no  mere 
change  of  temperature,  or  in  its  relations  to  latent  heat,  will  con- 
vert atmospheric  air  into  a  liquid  or  solid  form.  It  is  indeed  said 
that  intense  pressure  will  reduce  it  to  a  liquid  state,  and  we  know 
that  its  chemical  constituents  do,  when  in  combination,  become 


343  CHARACTERS  OJ  [JBoO/C    V. 

both  liquid  and  solid.  Still  the  difference  in  this  respect,  is  so  mark- 
ed between  the  air  of  our  atmosphere  and  another  class  of  elastic 
fluids,  that  we  are  not  warranted,  in  treating  of  its  mechanics,  to 
refuse  to  receive  the  permanence  of  its  elasticity,  as  one  of  its 
principal  characteristic  properties. 

The  researches  of  chemistry  have  shown  us  that  the  air  of  our 
atmosphere  is  not  homogeneous  in  its  chemical  constitution,  but 
that  it  is  a  mixture  of  several  fluids  of  the  same  mechanical  na- 
ture, differing  in  their  chemical  properties.  So,  also,  have  a  num- 
ber of  other  fluids  permanently  elastic  in  character,  that  form  no 
perceptible  portion  of  the  atmosphere,  been  discovered  ;  to  the 
whole  of  these  we  give  the  common  epithet  of  Gas. 

A  gas,  then,  is  a  material  substance,  belonging  to  the  class  of 
fluids,  gravitating  or  possessed  of  weight,  and  permanently  elas- 
tic. 

353.  When  water  is  heated  in  a  close  vessel,  an  elastic  fluid  is 
generated  ;  this  finally  acquires  such  expansive  force  as  to  break 
the  vessel  :  if  boiled  in  a  glass  vessel,  with  an  open  neck,  the 
space,  not  occupied  by  water,  will  appear  perfectly  transparent 
and  colourless  ;  but  a  cloud  will  appear  to  issue  from  the  neck. 
This  cloud,  if  examined,  will  be  found  to  be  composed  of  water 
in  a  state  of  minute  division,  and  we  infer  that  the  space  within 
the  vessel  is  filled  with  an  elastic  fluid.  To  render  this  more  evi- 
dent, adapt  a  syringe  to  the  neck  of  the  vessel,  containing  a  pis- 
ton fitted  air  tight,  and  depressed  to  the  bottom  of  the  syringe  ; 
when  the  water  begins  to  boil,  the  piston  will  begin  to  rise,  and 
will  speedily  reach  the  top  of  the  syringe;  if  cold  water  be  now 
applied  to  the  exterior  of  the  syringe,  the  piston  will  be  caused  sud- 
denly to  descend  ;  we  infer,  therefore,  that  the  elastic  fluid,  that 
had  been  generated  is  now  condensed.  The  visible  cloud  that  is- 
sues from  a  vessel,  was  formerly  called  vapour  or  steam  ;  we  now 
apply  those  terms  to  the  invisible  elastic  fluid  ;  and  we  distin- 
guish it  from  gases,  as  being  condensible  by  physicial  means, 
which  they  are  not. 

Numerous  experiments  show,  that  water  is  not  the  only  sub- 
stance, that  is,  when  heated,  converted  into  an  elastic  fluid  ;  few 
or  no  substances  indeed  exist,  that  do  not  become  volatile  at  a 
greater  or  less  degree  of  temperature,  and  all  these  volatilized 
matters  are  condensed  when  the  heat  is  withdrawn.  The  name 
of  vapour  is  hence  applied  to  them  all. 

Vapour,  then,  is  a  material  substance,  existing  in  the  fluid  form, 
elastic,  but  not  permanently  so,  being  capable  of  condensation  by 
cold. 

We  therefore  subdivide  elastic  fluids  into  two  classes  :  Gases, 
or  permanently  clastic  fluids  ;  and  Vapours,  or  condensible  elas- 


Book 


ELASTIC   FLUIDS.  349 


tic  fluids.  Physically  speaking,  the  line  that  separates  these  two 
classes,  is  not  distinctly  marked.  There  are  some  substances 
classed  as  gases,  that  are  condensible  with  no  great  difficulty; 
thus  ammonia  becomes  liquid  at  very  low  temperatures  ;  in- 
deed, few  or  none  of  the  gases  are  capable  of  retaining  their  elas- 
tic form  under  intense  pressure  ;  for  even  atmospheric  air  has, 
as  stated  by  Perkins,  been  reduced  to  the  liquid  form.  On  the 
other  hand,  there  are  some  vapours  that,  under  physical  or  me- 
chanical circumstances,  differing  in  no  great  degree  from  those 
we  find  existing  at  the  surface  of  the  earth,  would  appear  perma- 
nently elastic  :  thus,  the  vapour  of  ether  is  formed  under  ordi- 
nary circumstances  at  98°,  and  would,  in  a  climate  of  that  tem- 
perature, be  gaseous. 

The  elastic  force  with  which  gases  or  vapours  tend  to  expand 
themselves,  is  called  their  Tension. 

354.  The  tension  of  elastic  fluids  is  increased  by  heat,  and  di- 
minished by  cold  ;  thus,  under  a  given  pressure,  a  given  weight 
of  atmospheric  air,  and  in  general,  of  any  elastic  fluid,  is  found, 
when  heated,  to  occupy  a  greater  space.     This  may  be  simply 
shown^by  enclosing  a  portion  of  air  in  a  bladder,  and  exposing  it 
to  heat;  the  bladder  will  be  distended  and  finally  burst.     If  the 
bladder  be  completely  filled,  and  then  cooled,  it  will  cease  to  be 
distended,  and  the  distention  may  be  restored  by  raising  it  again 
to  its  original  temperature. 

355.  The   mass  that  composes  the  atmosphere  of  our  earth 
being  a  fluid,  the  general  properties  of  that  class  of  bodies  would 
lead  us  to  infer,  that  it  will  tend,  from  the  freedom  of  motion 
among  its  particles,  to  distribute  itself  uniformly  over  the  whole 
surface  of  the  earth.     This  is  confirmed  by  experience,  for  we 
find  the  atmosphere  exerting  a  pressure,  that  taken  at  a  mean,  is 
constant  at  every  point  upon  the  mean  surface  of  the  earth. 

This  property  is  not  confined  to  the  mass,  but  is  inherent  in 
each  of  its  several  chemical  constituents.  We  learri  this  from 
the  researches  of  Dalton,  who  has  shown  conclusively  that  every 
different  elastic  fluid  tends  to  distribute  itself  uniformly  over  the 
surface  of  the  earth,  and  thus  to  form  a  separate  atmosphere  of 
itself.  This  tendency  is  exerted  precisely  as  if  no  other  elastic 
fluid  were  present  ;  but  the  rapidity  with  which  the  distribution 
takes  place,  is  affected  by  the  fluid  resistance  of  the  other  elastic 
fluids  :  and  we  shall  see,  hereafter,  that  in  the  present  physical 
state  of  our  planet,  the  several  elastic  fluids  that  make  up  our 
atmosphere,  are  constantly  tending  to  a  state  of  equilibrium,  that 
they  never  completely  attain  This  discovery  of  Dalton,  we 
shall  find  of  great  value  ;  for  the  present,  we  shall  merely  state, 
that  it  has  fully  explained  the  fact,  observed  by  chemists,  but  not 


350  PRESSURE  OF  [Book  V. 

to  be  accounted  for  upon  the  principles  of  that  science  alone,  of 
the  constant  proportion  of  the  two  gases  that  compose  the  prin- 
cipal part  of  atmospheric  air. 

356.  The  fluid  pressure  of  the  atmosphere  was  first  exhibited 
by  Torricelli.  It  having  been  found  that  the  height  to  which 
water  rises  in  a  common  pump,  was  limited  ;  and  it  being  infer- 
red by  him  that  the  cause  was  mechanical,  he  saw  that  this, 
whatever  were  its  nature,  must  on  the  principle  of  §  326,  raise 
a  mass  of  denser  fluid  to  a  less  height.  He,  therefore,  inferred 
that  the  experiment  of  the  pump  might  be  performed  with  an  ap- 
paratus of  less^size,  provided  he  used  mercury  instead  of  water. 
Taking,  therefore,  a  glass  tube  of  about  three  feet  in  length,  he 
adapted  a  piston  to  it,  and  plunged  the  end  in  a  basin  of  mer- 
cury. On  drawing  up  the  piston,  the  mercury  followed  until  it 
reached  the  height  of  SO  inches,  at  which  the  fluid  ceased  to 
rise  ;  on  drawing  up  the  piston  farther,  a  space  was  left  between 
it  and  the  surface  of  the  mercury.  Now  as  the  limit  to  which 
water  rises,  in  a  well-constructed  common  pump,  is  34  feet;  and 
as  this  height  is  to  30  inches  in  the  inverse  ratio  of  their  respect- 
ive densities,  (See  Table  of  Specific  Gravities,  §  346),  the  pres- 
sures of  these  two  columns  of  the  liquids  are  equal,  and  require 
an  equal  force  for  their  support,  if  the  force  be  not  identical. 
Torricelli  at  once,  and  as  was  considered  at  the  time,  with  great 
boldness,  inferred  that  the  forces  which  supported  the  water  in 
the  pump,  and  the  mercury  in  the  tube,  were  identical,  and  that 
it  was  to  be  sought  in  the  fluid  pressure  of  the  atmosphere.  As 
the  weight  of  the  atmosphere  was  then  unknown,  and  the  fact  of 
its  compressibility  not  understood,  the  boldness  of  his  inference 
staggered  even  some  of  the  most  enlightened  of  his  cotempora- 
ries. 

Toricelli  also  planned  a  more  convenient  mode  of  performing 
his  experiment.  Taking  a  tube  of  the  same  length  as  before,  he 
closed  it  at  one  end,  and  filled  it  with  mercury  ;  placing  the 
finger  upon  the  open  end  of  the  tube,  he  inverted  it  in  a  basin 
of  the  same  liquid;  on  removing  the  finger,  the  mercury  sub- 
sided from  the  sealed  end  of  the  tube,  until  it  reached  the  height 
of  about  30  inches  above  the  level  of  the  mercury  in  the  basin, 
at  which  it  remained  stationary. 

That  the  cause  which  supports  a  column  of  water,  is  identical 
with  that  which  sustains  the  column  of  mercury,  in  these  expe- 
riments, may  be  shown,  by  pouring  water  upon  the  surface  of 
the  mercury  in  the  basin.  If  the  tube  be  now  raised  until  its 
open  end  be  above  the  level  of  the  mercury,  but  remain  still  im- 
mersed in  the  water,  the  mercury  in  the  tube  will,  from  its  su- 
perior weight,  fall  through  the  water  beneath  it;  the  water  will 


Book   J7".]  THE  ATMOSPHERE.  351 

enter  the  tube  to  supply  its  place,  but  so  far  from  ceasing  to  rise 
at  a  height  of  30  inches,  rushes  to  the  top  of  the  tube,  and  would 
fill  it,  even  were  its  length  as  great  as  34  feet. 

Pascal,  who  did  not  at  first  admit  the  conclusions  of  Torricelli, 
planned  an  experiment  v/hich  furnished  complete  evidence  of 
the  accuracy  of  the  views  of  that  philosopher.  It  has  been  shown, 
§  316,  that  that  the  pressure  of  a  fluid,  upon  a  given  base,  is  a 
function  of  the  depth.  If  then  the  cause  of  the  rise  of  the  mer- 
cury be  the  pressure  of  the  atmosphere,  it  is  obvious  that  on  car- 
rying the  Torricellian  apparatus  to  a  greater  height  above  the 
level  of  the  sea,  the  pressure  must  diminish,  and  the  column  of 
mercury  that  it  will  support  will  be  lessened  in  height.  This  ex- 
periment being  performed  at  the  base,  and  on  the  summit  of  the 
mountain  Puy  dt  Dome,  in  Auvergne  in  France,  gave  this  anti- 
cipated result;  a  diminution  in  the  height  of  the  column  of  mer- 
cury being  found  amounting  to  about  3  inches,  for  a  change  of 
level  of  3<500  feet. 

357.  The  apparatus  of  Torricelli,  then,  demonstrates  conclu- 
sively the  existence  of  atmospheric  pressure.     It  also  enables  us 
to  measure  its  amount.     For  the  pressure  of  the  column  of  mer- 
cury, with  which  that  of  the  atmosphere  must  be  in  equilibrio,  is, 
§  331,  equal  to  the  weight  of  a  prism  of  the  fluid  whose  height  is 
30  inches.     Upon  a  square  inch,  the  bulk  of  this  prism  is  30 
cubic  inches  ;  and  this  quantity  of  mercury  has  a  weight  that  dif- 
fers but  little  from  15  Ibs.    The  air  of  the  atmosphere,  therefore, 
presses,  at  the  level  of  the  sea,  with  a  force  equivalent  to  15  Ibs. 
upon  each  square  inch  of  surface.     This  quantity  of  15  Ibs.  per 
square  inch,  constitutes  that  unit  in  which  the  pressures  and  ten- 
sions of  elastic  fluids  are  measured,  and  which  is  called  an  Atmos- 
phere. 

Were  atmospheric  air  a  fluid  of  uniform  density  throughout, 
its  height  above  the  mean  surface  of  the  earth  might  be  calculated 
from  the  Torricellian  experiment ;  for  if  we  call  the  density  of 
water  1000,  that  of  mercury  is  13600  ;  and  that  of  air  at  a  mean, 
at  the  level  of  the  sea  is  usually  estimated  at  If  ;  the  mean  height 
of  the  column  of  the  mercury  in  the  barometer,  at  the  same 
level,  is  thirty  inches,  or  2i  feet.  Calculated  from  these  data,  the 
height  of  an  atmosphere  of  uniform  density  is  27600  feet. 

The  Torricellian  apparatus,  with  additions  and  under  modifi- 
cations that  will  be  hereafter  described,  goes  at  present  by  the 
name  of  the  Barometer. 

358.  The  common  pump,  to  an  observation  on  the  action  of 
which  the  Torricellian  experiment  was  due,  is  an  apparatus  whose 


359 


PRESSUHE  OF 


[Book  V. 


A' 


form  and  mode  of  action  may  be  understood  from  the  annexed 
figure.  AA',  is  a  pipe  of  any  convenient  length, 
less  than  the  limit  to  which  water  may  be  raised  by 
atmospheric  pressure  ;  BB',  a  barrel  generally  of 
greater  diameter  than  the  pipe ;  c,  a  valve  open- 
ing upwards,  at  the  junction  of  the  pipe  and  barrel ; 
d,  a  piston,  moveable  by  means  of  the  rod  e ;  in 
this  piston  there  is  also  a  valve  opening  up- 
wards. When  the  piston  is  raised,  the  air  in  the 
barrel  between  the  two  valves  is  expanded,  and 
its  tension  diminished  ;  the  air  in  the  barrel,  there- 
fore, opens  the  valve  c,  and  the  whole  mass  of  air 
tends  to  become  less  dense  ;  but  as  this  pipe  com- 
municates with  water  in  a  basin,  that  is  pressed  at 
its  surface  by  the  air  of  the  atmosphere,  this  pres- 
sure causes  the  water  to  rise  in  the  tube,  until  the 
tension  of  this  confined  air  becomes  equal  to  the 
pressure  of  the  atmosphere.  On  depressing  the 
piston  dy  the  valve  in  it  opens,  and  air  passes  upwards  from  the 
barrel  as  the  former  descends  ;  but  the  valve  c[is  closed  by  the 
downward  pressure,  and  the  volume  of  water  that  has  entered  the 
pipe,  remains.  On  again  raising  the  piston,  the  same  action  takes 
place  as  at  first,  and  an  additional  quantity  of  water  enters  the 
pipe  :  a  second  depression  of  the  piston  causes  the  same  effects  as 
the  first.  Thus  a  column  of  water  will  be  raised  by  the  alter- 
nating motion  of  the  piston,  until  that  fluid  reaches  the  piston  in 
its  lower  position.  On  raising  the  piston  when  the  water  has 
reached  it,  that  fluid  will  be  compelled  to  follow  it  by  the  pres- 
sure of  the  atmosphere  ;  when  the  piston  is  again  depressed,  the 
water  flows  through  the  valve  situated  in  it  ;  and  on  its  being 
again  raised,  the  water  will  be  lifted  upon  its  surface  until  it 
reaches  and  flows  out  of  the  spout  F. 

Although  in  theory  the  limit  of  the  height  to  which  water  may 
be  raised  by  the  common  pump,  measured  from  the  surface  in 
the  basin  beneath,  to  the  highest  position  of  the  moveable  piston, 
is  34  feet ;  it  is  not  found  practicable,  with  pumps  of  ordinary 
structure,  to  raise  that  liquid  more  than  about  28  feet.  This  dif- 
ference arises  from  the  difficulty  of  making  the  apparatus  abso- 
lutely air-tight. 

Liquids  are  supported  in  a  syphon,  upon  the  same  principle 
that  they  are  raised  in  the  common  pump.  If  a  bent  tube  be  filled 
with  a  liquid,  and  inverted,  without  permitting  the  liquid  to 


escape. 


and  one  of  its  branches  be  immersed  in  a  vessel  contain- 


ing a  mass  of  the  same  liquid,  so  much  of  the  liquid  in  the  tube  as 
is  above  the  level  plane,  of  which  the  surface  of  that  in  the  vessel 


Book  VJ\  ELASTIC  FLUIDS.  353 

forms  a  portion,  will  be  supported  by  the  pressure  of  the  atmos- 
phere ;  and  the  columns,  in  the  two  branches  of  the  tube,  lying 
above  this  level,  exactly  balance  each  other ;  but  the  remainder 
of  the  column,  in  the  branch  that  is  not  immersed,  will  cause  the 
column  of  which  it  is  a  part,  to  preponderate  and  descend  ;  the 
pressure  of  the  atmosphere  will  cause  an  equal  quantity  of  liquid 
to  rise  in  the  opposite  branch  to  supply  its  place,  and  thus  a  con- 
tinual stream  will  flow  from  the  open  end  of  the  tube,  until  the 
level  of  the  liquid  in  the  vessel  descends  as  low  as  the  orifice  on 
the  branch  of  the  tube  that  is  not  immersed. 

The  apparatus  called  the  Syphon,  is  a  bent  tube,  and  has  usu- 
ally branches  of  unequal  length,  the  shorter  of  which  is  immersed 
in  the  vessel.  Hence  the  flow  of  liquid  will  continue  until  its 
level  descends  as  low  as  the  opening  in  the  immersed  branch  ; 
the  air  will  then  enter  the  syphon,  and  the  whole  of  the  liquid 
will  be  discharged  from  it. 

Instead  of  filling  the  syphon,  and  inverting  it,  the  air  may  be 
extracted  by  a  pump,  or  by  the  action  of  the  lungs  ;  the  mouth 
is,  in  the  latter  case,  applied  to  a  pipe  adapted  for  the  purpose, 
and  the  end  of  the  syphon  that  is  not  immersed,  closed  by  the 
finger,  or  by  a  stopcock.  The  apparatus,  in  this  form,  is  called 
a  Crane,  and  is  used  for  the  purpose  of  decanting  liquids.  The 
flow  of  liquid  from  a  syphon,  will  have  a  retarded  velocity,  pro- 
vided the  vessel  be  permitted  to  empty  itself.  But  if  it  be  kept 
full,  by  any  appropriate  means,  the  velocity  remains  constant. 


45 


354  OF  THE  AIK  PUMP.  [Book  V. 

CHAPTER  VI. 

OP  THE  AIR  PUMP. 

359.  The  mechanical  properties  of  the  air  may  be  best  inves- 
tigated experimentally,  by  means  of  the  apparatus  called  the  Air 
Pump. 

If  an  instrument  similar  in  construction  to  a  common  pump,  but 
of  more  accurate  workmanship,  be  adapted  to  a  tight  vessel,  the 
action,  which  in  the  common  pump  diminishes  the  tension  of  the 
air  in  such  a  manner  as  to  allow  water  to  be  forced  up  by  the  pro- 
sure  of  the  atmosphere,  will  now  act  merely  to  rarefy  the  air  con- 
tained in  the  vessel.  When  the  piston  is  raised,  the  air  in  the 
vessel  expands  in  consequence  of  its  elasticity,  opens  the  lower 
valve,  and  fills  the  whole  space  beneath  the  piston  with  air  of  an 
uniform  tension,  but  of  diminished  density.  When  the  piston  is 
depressed,  the  air  contained  between  it,  and  the  lower  valve,  is 
compressed  until  it  reach  the  same  density  as  the  external  atmos- 
phere ;  it  then  opens  the  upper  valve  and  passes  out.  On  raising 
the  piston  again,  a  farther  rarefaction  takes  place  ;  and  thus,  at 
each  alternate  motion  of  the  piston  a  new  expansion  takes  place, 
and  a  portion  of  the  air  originally  contained  in  the  vessel,  passes 
out. 

We  call  the  space  within  the  receiver,  from  which  the  air  has 
been  extracted,  the  Vacuum  of  the  Air  Pump.  This  is  less  per- 
fect than  that  at  the  top  of  the  barometer,  which  is  called  the  Tor- 
ricellian Vacuum. 

The  first  air  pump  was  invented  by  Otto  Guericke,  of  the  city 
of  Magdeburg,  and  had  the  form  we  have  used  for  our  illustration. 
The  joint  between  the  pump  and  the  close  vessel,  was  rendered 
air  tight  by  a  collar  of  leather,  moistened  with  water.  We  now 
use  in  the  several  joints  of  the  pump,  leather  dipped  in  a  liquid 
oleaginous  matter. 

The  air  pump  has  received  many  modifications  in  form,  and 
improvements  in  structure,  since  the  time  of  its  invention,  of 
which  the  following  are  among  the  most  important. 

A  close  vessel  is  ill  suited  for  the  purpose  of  introducing  sub- 
stances, or  apparatus  for  experiment,  into  the  vacuum  of  the  air 
pump  ;  but  if  an  open  receiver  be  taken,  and  placed  upon  a  plate, 
with  an  orifice  in  the  middle  of  which  the  pump  communicates 
by  a  pipe ;  when  the  pump  is  set  in  action,  it  will  tend  to  rarefy 
the  air  in  the  receiver,  and  the  pressure  of  the  atmosphere  will 
fix  the  latter  to  the  plate  of  the  pump.  The  receiver  will  then. 


Book  F.]  OP  THE  AIR  PUMP.  355 

if  the  joints  between  it  and  the  plate  be  airtight,  be  to  all  intents 
and  puposes  a  close  vessel.  To  render  this  joint  air  tight,  the 
plate  was  at  first  covered  with  a  piece  of  oiled  leather.  It  was 
however  found,  that  the  evaporation  of  the  oil  from  the  leather, 
lessened  the  extent  of  rarefaction,  by  continually  supplying  a  fresh 
elastic  fluid.  For  this  reason,  then,  the  plate  of  the  pump  (usu- 
ally made  of  brass)  has  been  ground;  the  rim  of  the  receiver  is 
also  ground,  in  order  to  adapt  itself  to  the  plate.  In  this  way  the 
joint  will  frequently  be  air  tight,  without  any  other  precaution, 
but  more  generally  it  is  necessary  to  touch  the  rim  of  the  receiver 
with  a  little  oil. 

As  oil  is  acted  upon  by  brass,  and  corrodes  it,  glass  plates  have 
been  substituted  for  brass,  and  are  ground  in  the  same  manner. 

The  working  of  the  piston  of  the  pump,  is  opposed  by  the  pres- 
sure of  the  atmosphere  ;  hence  another  pump  barrel  has  been  add- 
ed, and  the  two  pistons  made  to  act  alternately,  the  one  rising 
as  the  other  falls.  By  this  method,  the  pressures  on  the  pistons 
are  made  to  oppose  each  other.  To  work  the  pistons  with  greater 
facility,  their  rods  are  cut  into  teeth,  forming  racks  :  between 
these,  and  interlocking  with  both,  is  placed  a  pinion,  or  toothed 
wheel.  The  latter  is  turned  by  a  winch,  and  sometimes  by  a 
double  handle. 

The  resistance  of  the  atmosphere  may  be  in  a  great  degree  re- 
moved, by  making  the  cylinder  of  the  pump  air  tight ;  the  piston 
rod  must,  in  this  case,  work  through  a  collar  of  leathers  ;  a  late- 
ral spout  is  therefore  placed  to  permit  the  escape  of  the  air,  and  this 
contains  a  third  valve,  opening  upwards.  A  pump  of  this  struc- 
ture works  with  greater  ease,  as  the  exhaustion  of  the  receiver 
increases,  while  in  those  with  open  barrels,  the  resistance  aug- 
ments with  the  exhaustion. 

In  the  earlier  pumps,  both  the  valves  were  opened  by  the  elas- 
ticity of  the  air  ;  and  thus  a  limit  to  the  exhaustion  was  attained, 
in  the  resistance  of  the  valve  to  an  effort  to  open  it.  In  order  to 
increase  the  power  of  the  pump  in  exhaustion,  the  lower  valve  is 
now  made  to  open  by  the  rise  of  the  piston,  and  is  shut  by  its  de- 
scent. 

Such  an  arrangement  is  represented  on  the  succeeding  page, 
being  a  section  of  the  air  pump  of  Demoutier. 

P  P'  is  the  piston  in  which  is  seen  the  upper  valve  s  ;  the  lower 
valve  is  represented  at  c;  it  has  a  conical  form,  and  is  attached  to 


356 


01   THE  AIR  PUMP. 


[Book  V. 


a  rod  cty  that  passes  through  the  piston,  and  its  leather  packing, 
with  so  much  friction  as  to  be  air  tight.  As  soon  as  the  piston 
begins  to  rise,  the  rod  ct  will,  in  consequence  of  the  friction, 
move  with  the  piston,  and  the  valve  c  opens  ;  but  the  rod  moves 
through  no  more  space  than  is  just  necessary  to  open  the  valve, 
being  checked  by  a  shoulder  at  r,  that  strikes  against  a  plate  that 
forms  a  lid  to  the  barrel  of  the  pump. 

The  valves  of  the  pumps  that  first  succeeded  that  of  Guericke, 
were  made  of  strips  of  bladder,  or  of  oiled  silk  :  in  the  pump  just 
represented,  the  upper  valve  is  of  steel,  resting  on  an  oiled  leather 
seat;  the  lower  valve  is  a  leather  conic  frustum,  applying  itself  to 
a  hollow  frustum  formed  in  a  steel  plate.  In  the  best  English 
pumps,  the  valves  and  seats  are  both  of  metal,  and  the  form  that 
of  a  conic  frustum. 

We  have  stated  that  oil  and  brass  mutually  act  upon  each  other; 
hence,  the  barrels  and  pistons  that  were  originally  made  of  brass, 
and  rendered  air  tight,  by  packing  of  leather  soaked  in  oil,  were 
rapidly  worn.  To  remedy  this  defect,  the  pistons  are  now  made 
of  steel,  which  is  effectually  preserved  from  rust  by  the  oil  used 
in  the  packing,  and  the  barrels  of  the  best  pumps  are  made  of 
glass. 

360.  The  power  of  air  pumps,  to  exhaust  the  receivers,  placed 
upon  their  plates,  is  measured  by  means  of  instruments  called 
Gauges.  These  are  of  several  kinds. 

(1).  The  Barometer  Gauge. — This  consists  of  a  glass  tube  about 
30  inches  in  length,  and  open  at  both  ends  ;  it  is  placed  in  a  ver- 
tical position  attached  to  the  stand  of  the  pump  ;  the  lower  end 
is  plunged  in  a  basin  of  mercury  ;  the  upper  is  cemented  air  tight 


Book  K]  OP  THE  AIR  PUMP.  357 

to  a  pipe  that  leads  to  the  opening  in  the  plate  of  the  pump,  by 
which  the  air  passes  from  the  receiver ;  or  it  may  communicate 
in  some  other  manner  with  the  valves  of  the  pump.  As  the  air 
in  the  receiver  is  exhausted,  the  pressure  of  the  external  atmos- 
phere, forces  the  mercury  in  the  basin  to  rise  in  the  glass  tube. 
The  difference  between  the  height  to  which  the  mercury  is  thus 
raised,  and  that  at  which  it  is  supported  at  the  same  time  in  the 
sealed  Torricellian  tube,  or  Barometer,  is  obviously  a  measure  of 
tension  of  the  air  yet  remaining  in  the  receiver.  We  shall  here- 
after see  that  this  tension  is  exactly  proportioned  to  the  quantity 
of  air  that  remains. 

(2).  A  tube  6  or  S  inches  in  length,  sealed  at  one  end,  and 
filled  with  mercury,  is  inverted  in  a  basin  of  the  same  liquid. 
The  mercury  is  of  course  supported  by  the  pressure  of  the  atmos- 
phere, and  continues  to  fill  the  tube.  This  apparatus  is  placed 
beneath  a  receiver,  communicating  with  the  valves  of  the  pump. 
During  the  first  stages  of  the  exhaustion,  the  mercury  still  remains 
supported,  but  so  soon  as  the  tension  oT  the  contained  air  becomes 
less  than  is  sufficient  to  support  such  a  z J.um-;  of  mercury,  that 
liquid  begins  to  fall  in  the  tube  ;  the  height  at  which  it  stands 
above  the  level  of  that  in  the  basin,  is  a  rne'.'-ure  of  the  tension 
of  the  remaining  air.  This  is  called  the  Short  A-aromete.*  Gauge. 

(3).  The  Syphon  Gauge  acts  upon  the  same  >nncipl3  as  the 
last;  but  instead  of  plunging  the  open  end  of :  .3  tube  in  a  basin 
of  mercury,  it  is  bent  upwards  ;  the  whole  is  their  fare  composed 
of  two  parallel  branches,  one  open,  and  the  other  closed.  The 
latter  is  filled,  before  the  action  of  the  pump  begir  s,  with  mer- 
cury. 

These  two  gauges  are  placed  beneath  separate  receivers,  ex- 
pressly adapted  for  them  to  an  additional  part  of  the  pump,  in 
order  that  they  may  be  used  at  the  same  time  that  other  receivers 
are  placed  upon  the  principal  plate. 

(4).  The  Pear-Gauge  —  a  vessel  of  glass  of  the  shape  denoted 
by  its  name,  having  a  small  opening  at  the  larger  end,  is  sunk 
by  means  of  weights  into  a  basin  of  mercury,  with  its  opening 
downwards.  This  apparatus  being  placed  beneath  a  receiver,  and 
the  pump  set  in  action,  the  air  contained  in  the  pear-shaped  vessel 
will  expand,  and  make  its  way  through  the  mercury  :  thus,  the 
exhaustion  withki  it  will  correspond  with  that  effected  in  the 
receiver.  On  re-admitting  the  air  to  the  receiver,  the  pressure 
upon  the  mercury  in  the  basin  will  force  it  into  the  pear-shaped 
vessel,  until  the  portion  of  air  left  within  it  be  restored  to  about 
its  original  density.  Hence,  th-  relation  between  the  part  of  the 
vessel  not  filled  by  the  mercury,  and  its  whole  capacity,  will  be  a 
measure  of  the  exhaustion  of  the  pump. 


358  OP  THE  AIR  PUMP.  [Book  V. 

When  this  gauge  is  used  with  a  pump,  to  whose  plate  the  re- 
ceiver is  adapted  by  means  of  a  sheet  of  leather  soaked  in  oil,  it 
will  exhibit  a  far  higher  degree  of  exhaustion  than  any  of  the  three 
preceding  gauges.  .  This  grows  out  of  the  fact  that  the  elastic  va- 
pour of  the  oil,  which  affects  them,  cannot  enter  it.  This  obser- 
vation led  to  the  improvement  we  have  mentioned,  of  fitting  the 
receivers  to  the  plate  of  the  pump  by  grinding. 

361.  The  air  pump  affords  a  great  variety  of  proofs  of  the  pres- 
sure of  the  atmosphere,  of  which  a  few  may  be  cited  : 

(1).  In  the  manner  in  which  the  receiver  is  fixed  to  the  plate 
of  the  pump,  in  such  a  way  as  to  act  as  a  close  vessel ;  in  its  re- 
maining firmly  attached  to  the  plate  after  the  air  is  exhausted. 

(2).  The  hand  placed  upon  an  open  receiver,  will  in  like  man- 
ner be  firmly  fastened  to  it ;  and  the  pressure  which  was  before 
equal  on  both  sides  of  the  hand,  and  therefore  imperceptible,  will 
become  painful. 

(3).  A  square  flask  of  glass,  screwed  to  the  opening  in  the 
plate  of  the  pump,  will  be  broken,  as  will  a  glass  plate,  ground 
to  fit  an  open  receiver. 

(4).  A  small  receiver,  suspended  by  means  of  a  rod  passing 
through  a  collar  of  leathers,  within  a  larger  receiver,  may  be  de- 
pressed by  means  of  the  rod,  till  it  rests  on  the  plate  of  the  pump. 
If  this  small  receiver  do  not  cover  the  orifice  by  which  the  plate 
communicates  with  the  valves,  the  re-admission  of  air  into  the 
larger  receiver,  fixes  the  smaller  to  the  plate  of  the  pump;  nor  can 
it  be  removed  until  the  air  be  again  exhausted. 

(5).  Two  hemispheres  of  brass  ground  to  fit  each  other,  may 
be  affixed  to  the  plate  of  the  pump  by  means  of  a  pipe  proceeding 
from  one  of  them,  and  which  is  screwed  to  the  orifice.  If  the 
air  be  exhausted,  they  will  be  found  to  be  pressed  powerfully 
together.  If  there  be  a  stop-cock  on  the  pipe  of  communication, 
it  may  be  closed,  and  the  apparatus  detached.  The  spheres  held 
together  by  the  pressure  of  the  atmosphere  may  be  then  separated 
by  means  of  weights  ;  one  of  them  being  suspended,  and  the 
weights  attached  to  the  other.  The  quantity  of  weight  required  to 
separate  them,  will  be  a  measure  of  the  pressure  they  sustain  :  as 
the  surface  of  a  sphere  of  known  diameter  can  be  calculated,  this 
was  used  as  a  measure  of  the  atmosphere's  pressure  upon  each  unit 
of  square  measure.  It  is,  however,  affected  by  the  air  which 
remains  in  the  apparatus,  and  which  the  pump  cannot  wholly  ex- 
haust :  it  is,  therefore,  inferior  in  accuracy  to  the  tube  of  Torricelli. 

Experiments  for  illustrating  the  pressure  of  the  atmosphere  by 
means  of  the  air  pump,  may  be  multiplied  to  a  great  extent,  but 
the  above  are  sufficient  for  our  purpose. 


Book  P.]  OF  THE  AIR  PUMP.  359 

362.  That  the  pressure  of  the  atmosphere  is  the  cause  of  the 
rise  of  fluids  in  pumps,  and  of  mercury  in  the  Torricellian  ex- 
periment, may  also  be  shown  by  means  of  the  air-pump.    Thus, 
if  a  model  of  a  pump  be  passed  through  a  collar  of  leathers,  adapt- 
ed to  the  top  of  a  receiver,  it  will  be  found  to  be  without  effect, 
when  the  reservoir  in  which  it  is  plunged  is  not  pressed  by  air. 
So  if  a  barometer  be  in  like  manner  inserted  into  the  top  of  a  re- 
ceiver, the  mercury  will  fall  in  the  tube  as  the  air  is  exhausted 
from  the  receiver. 

363.  To  show  the  weight  of  air,  a  flask  furnished  with  a  stop- 
cock is  weighed ;  it  is  then  screwed  to  the  plate  of  the  pump, 
and  the  air  exhausted ;  the  stopcock  being  closed,  it  may  be  re- 
moved without  admitting  air,  and  being  again  weighed,  it  will  be 
found  lighter  than  before.  To  obtain  the  exact  weight  of  a  given 
bulk  of  air,  demands  certain  precautions  which  will  be  hereafter 
stated. 

364.  The  elasticity  of  the  air  is  manifest  from  the  manner  in 
which  the  air-pump  itself  works ;    it  may  also  be  exhibited  in  va- 
rious other  manners,  thus  : 

(1.)  If  a  bladder,  partially  inflated  with  air,  be  placed  under 
a  receiver,  and  the  air  withdrawn  from  the  latter,  the  bladder 
swells,  and  is  distended;  on  re-admitting  the  air  to  the  receiver, 
the  bladder  collapses  to  its  original  dimensions. 

(2.)  If  the  bladder,  when  partially  filled,  be  loaded  with 
weights,  and  placed  beneath  a  receiver,  the  contained  air  will,  in 
expanding  itself  as  the  receiver  is  exhausted,  lift  the  weights. 

(3.)  If  a  glass  mattrass  be  inverted  in  a  vessel  containing  a 
liquid,  and  placed  beneath  the  receiver  of  an  air-pump ;  as  the 
air  is  exhausted  from  the  receiver,  that  confined  in  the  mattrass 
by  the  liquid  will  expand,  and  escape  in  visible  bubbles  through 
the  fluid  mass.  On  re-admitting  the  air,  the  liquid  will  be  forced 
up  the  neck  of  the  mattrass,  and  into  the  bulb,  which  it  will,  if 
the  exhaustion  have  been  sufficient,  nearly  fill.  On  a  second  ex- 
haustion, the  air  will  expand,  as  will  be  manifested  by  the  de- 
scent of  the  liquid  confined  in  the  mattrass;  and  the  latter  will  be 
forced  back  a  second  time  on  the  re-admission  of  air. 


360  EQUILIBRIUM  OF  [Book   V. 

CHAPTER  VII. 

EQUILIBRIUM  OF  PERMANENTLY  ELASTIC  FLUIDS. 

365.  Although  the  elasticity  of  the  air  is  demonstrated  by  the 
experiments  cited  at  the  close  of  the  last  chapter,  they  do  not 
exhibit  the  law  of  that  elasticity  ;  nor  do  they  point  out  what  re- 
lation the  force,  with  which  it  expands  itself,  bears  to  the  force 
by  which  it  is  compressed.  This  is  a  question  whose  importance 
demands  a  particular  investigation,  which  may  be  performed  by 
the  following  experiments. 

(1.)  Take  a  glass  vessel  with  a  narrow  neck,  (one  nearly  of  a 
spherical  figure  is  best  adapted  for  the  purpose),  and  pour  into  it 
a  small  portion  of  mercury ;  if  we  then  screw,  air-tight,  to  its 
neck  a  slender  tube  of  glass  open  at  both  ends,  of  the  length  of 
at  least  thirty  inches,  whose  lower  end  is  wholly  immersed  in  the 
mercury,  the  air  that  fills  the  upper  part  of  the  vessel  will  be 
wholly  separated  from  the  general  mass  of  the  atmosphere.  If 
this  apparatus  be  placed  beneath  a  tall  receiver,  upon  the  plate  of 
the  air  pump,  and  the  air  exhausted  ;  the  pressure  on  the  air, 
confined  in  the  vessel,  will  be  lessened  in  proportion  as  the  ten- 
sion of  the  air  in  the  receiver  is  diminished  ;  the  elastic  force  of 
the  confined  air  will,  in  consequence,  cause  a  column  of  mercury 
to  rise  in  the  tube,  and  the  height  of  this  column  will  be  a  mea- 
sure of  the  difference  between  the  tension  of  the  air  in  the  re- 
ceiver, and  that  of  the  air  of  its  original  density  confined  in  the 
vessel.  If  the  pump  have  a  long  barometer  gauge  (No.  1  of 
§  357),  the  mercury  that  rises  in  it  furnishes  a  similar  measure  of 
the  difference  between  the  pressure  of  the  external  atmosphere 
and  the  tension  of  the  air  remaining  in  the  receiver. 

The  relation  between  the  two  columns  of  mercury  furnishes  a 
direct  comparison  between  the  pressure  of  the  atmosphere,  and  the 
elasticity  of  the  air  that  composes  the  part  on  which  that  pressure 
takes  place;  for  the  confined  air  has  obviously  been  enclosed  un- 
der that  very  pressure.  On  comparing  the  heights  of  these  two 
columns,  during  all  the  periods  of  the  exhaustion,  they  will  be 
found  to  all  appearance  exactly  equal,  the  mercury  rising  in  the 
two  iubespaHpa*9U.  The  small  change  in  the  bulk  of  the  con- 
fined air,  growing  out  of  the  mercury  it  forces  out  of  the  vessel, 
is  insensible.  Hence,  it  is  obvious  that  the  elasticity  of  air, 
at  the  ordinary  density  of  the  atmosphere,  is  exactly  equal  to  the 
pressure. 


Book  VJ\  PERMANENTLY  ELASTIC  FLUIDS.  361 

(2.)  To  examine  the  effect  of  increased  pressure,  we  take  a  cy- 
lindrical glass  tube  bent  into  two  branches  ;  one  of  these  is  closed, 
and  the  other  open.  Mercury  is  then  poured  in  until  it  fill  the 
bend  of  the  tube  ;  a  portion  of  the  air  that  would  be  thus  confined 
in  the  closed  branch  is  permitted  to  escape,  by  inclining  the  ap- 
paratus, until  the  mercury  stands  at  the  same  level  in  bolh  branch- 
es of  the  tube.  The  confined  air  has  then  no  other  pressure  to 
sustain  than  that  of  the  atmosphere,  for  the  columns  of  mercury 
in  the  two  branches  counterbalance  each  other.  The  air,  there- 
fore, that  is  confined,  has,  as  may  be  inferred  from  the  preceding 
experiment,  the  same  density  as  that  of  the  adjacent  stratum  of 
the  atmosphere.  In  order  to  increase  the  pressure,  mercury  is 
poured  into  the  open  branch  of  the  tube  ;  the  confined  air  is  then 
affected  by  a  pressure  equal  to  the  sum  of  that  of  the  atmosphere, 
and  that  of  the  column  of  mercury  measured  from  the  level  at 
which  it  stands  in  the  close  branch  of  the  tube. 

In  this  experiment,  the  level  of  the  mercury  in  the  closed 
branch  of  the  tube  will  be  found  to  rise  as  the  pressure  caused  by 
the  column  in  the  open  branch  increases,  thus  marking  a  conden- 
sation in  the  enclosed  air.  By  the  uniform  result  of  all  experi- 
ments, after  employing  the  precautions  that  will  be  presently 
stated,  it  is  found  that  the  space  occupied  by  the  confined  air,  is 
inversely  as  the  pressure  to  which  it  is  subjected. 

In  order  to  express  this  fact,  let  B  be  the  original  volume  of 
the  air,  under  the  pressure,  /?,  namely  that  of  the  atmosphere  ;  B' 
a  volume  under  the  increased  pressure,  p'  ; 


whence  we  obtain 

B'=^;  (376) 

and  for  the  volume  B",  under  any  other  pressure  p", 

Ep 

"O/>  __          *      . 

'p'" 

dividing  bv  B'  we  obtain 


whence 

B"=-^l.  (376) 

P 

By  which  we  may  calculate  the  relation  of  the  volumes  of  the 
same  mass  of  air  under  any  given  pressures  whatsoever. 
The  precautions  to  which  we  have  referred,  grow  out  of  the 

following  circumstances  : 

46 


362  EQUILIBRIUM    OF  [Book    V. 

(a)  Air  in  being  com  pressed  has  its  temperature  raised  ;  hence 
the  apparatus  must  be  permitted  to  cool  down  to  the  temperature 
of  the  surrounding  air  before  the  observation  is  made. 

(b)  All  air  contains  a  greater  or  less  proportion  of  aqueous  va- 
pour, and  this,  as  we  shall  presently  see,  is  not  affected  by  pres- 
sure in  exactly  the  same  manner  as  air,  under  similar  circum- 
stances.    In  order  to  make  the  experiment  perfectly  satisfactory, 
the  tube  must  be  well  dried,  and  the  moisture  withdrawn  from 
the  air  it  contains,  by  exposing  the  latter  to  the  contact  of  hygro- 
metric  substances,  for  some  hours  before  the  experiment  is  made. 

The  space  being  found,  after  these  precautions  have  been  taken, 
to  be  inversely  proportioned  to  the  pressure,  it  may  be  inferred 
that  in  the  case  of  the  condensation  of  air,  such  as  is  usually  found 
at  the  surface  of  the  earth,  the  density  is  always  proportioned  to 
the  pressure  ;  and  to  this,  the  tension,  or  elastic  force,  is  equal. 

(3.)  It  remains  that  the  law  of  elasticity  should  be  determined 
for  air  of  diminished  density.  For  this  purpose,  take  the  Torri- 
cellian apparatus  and  fill  the  tube  partly  with  mercury  ;  the  re- 
maining part  will  continue  to  contain  air.  On  placing  the  fin- 
ger upon  the  open  end,  and  inverting  the  tube,  the  air  will 
rise  through  it  to  the  close  end,  and  will,  so  long  as  the  finger  is 
tightly  pressed  on  the  opening,  occupy  the  same  space  it  did  at 
first.  But  on  immersing  the  open  end  of  the  tube  in  a  basin  of 
mercury,  the  confined  air  is  no  longer  compressed  by  the  whole 
force  of  the  atmosphere,  for  the  latter  must  also  support  the  co- 
lumn of  mercury.  The  confined  air  then,  being  less  compressed, 
expands  itself,  and  causes  the  mercury  to  descend  ;  and  it  finally 
comes  to  rest  in  such  a  position  that  the  sum  of  the  pressure  of 
the  column  of  mercury,  and  of  the  tension  of  the  confined  air,  is 
equal  to  the  pressure  of  the  atmosphere. 

To  ascertain  the  law  from  this  experiment,  let  /be  the  tension 
of  the  air  after  its  expansion  ;  />,  the  pressure  of  the  atmosphere  ; 
6,  the  original  volume  of  the  air,  and  x,  that  to  which  it  expands 
itself;  it  is  found  that 


whence  we  obtain 

l=-r  (377) 

which  is  identical  with  the  formula  (375)  ;  hence  the  law  is  the 
same,  in  the  case  of  rarefied  as  in  that  of  condensed  air. 

The  same  experiments  may  be  performed  with  dry  gases,  and 
the  results  are  found  the  same  as  when  they  are  performed  with 
atmospheric  air.  Hence,  the  air  of  the  atmosphere  and  all  other 


Book 


PERMANENTLY  CLASTIC  FLUIDS.  363 


dry  gases,  at  constant  temperatures,  occupy  spaces  that  are  in- 
versely as  the  pressures  to  which  they  are  subjected.  This  law 
was  originally  discovered  by  Mariotte,  and  goes  by  his  name.  It 
is  true  at  all  mean  and  usual  pressures,  but  ceases  to  be  so  at  cer- 
tain limits.  Were  it  absolutely  true,  the  smallest  possible  quan- 
tity of  air  would,  on  the  pressure  being  wholly  removed,  occupy 
a  space  infinitely  great  ;  while  there  could  be  no  space  so  small 
into  which  the  largest  mass  of  the  air  could  not  be  compressed  by  a 
sufficient  force.  The  limit  is  found,  on  the  one  hand,  in  the  fact 
that  nearly  all  the  gases  have  been  condensed  into  the  liquid 
form  ;  and  even  atmospheric  air,  as  is  stated  by  Perkins,  has  been 
reduced  to  that  state.  The  limit,  on  the  other  hand,  appears  ra- 
ther to  arise  from  the  relation  of  expanding  air  to  heat  ;  for  in 
the  higher  regions  of  the  atmosphere,  the  expansion  produces  a 
cold  so  intense,  that  the  diminution  of  temperature  will  finally 
produce  an  effect  equal  and  contrary  to  that  caused  by  the  re- 
moval of  pressure.  Laplace,  in  stating  this  limit,  has  expressed 
the  opinion  :  that  although  the  existence  of  the  limit,  and  conse- 
quently the  finite  extent  of  the  atmosphere,  is  capable  of  demon- 
stration ;  the  exact  height  at  which  it  ceases  to  expand  is  not 
within  the  reach  of  calculation.  We  are  therefore  compelled  to 
have  recourse  to  a  physical  fact  to  estimate  the  probable  extent 
of  the  atmosphere  of  the  earth  ;  this  is  the  phenomenon  of  twi- 
light, whence  it  is  inferred,  that  the  air  of  the  atmosphere  still  re- 
tains a  sufficient  density  to  reflect  the  rays  of  light,  at  a  distance 
of  forty  miles  from  the  surface  of  the  earth. 

Were  atmospheric  air  capable  of  indefinite  expansion,  the  mass 
that  surrounds  the  earth  would  have  distributed  itself  around  the 
earth  and  moon,  in  the  ratio  of  their  respective  masses,  by  the 
influence  of  the  attraction  of  gravitation  ;  and  generally,  had  any 
one  planet,  at  the  time  of  the  creation,  been  surrounded  by  an  at- 
mosphere of  such  a  character,  all  would  have  derived  "from  it 
atmospheres  distributed  inasimilar  ratio.  Now  the  moon  has  no 
perceptible  atmosphere,  and  we  have  no  reason  to  conclude  that 
the  other  bodies  of  the  solar  system  have  atmospheres,  of  the 
density  and  mass  that  might  have  been  inferred  from  the  law  of 
Mariotte  :  hence,  again,  we  find  a  corroboration  of  the  opinion, 
that  the  atmosphere  of  the  earth  has  a  finite  extent.  The  same 
was  inferred  by  Wollaston,  from  astronomic  observations,  into 
the  detail  of  which  it  is  not  our  province  to  enter. 

366.  We  may  now  proceed  to  investigate  the  conditions  of 
equilibrium  in  the  air  that  composes  our  atmosphere,  under  the 
supposition  that  it  is  of  uniform  temperature  throughout. 


EQUILIBRIUM  OF  [Book    V. 

If  we  use  the  notation  of  §  316,  the  density  s  being,  by  the  law 
of  Mariotte,  directly  proportioned  to  the  pressure,  p, 

s=mp ;  (378) 

in  which  equation,  m  is  a  constant  co-efficient  to  be  determined 
by  experiment. 

The  equation  (352), 

dp=3  dz, 
will  give 

mz=log.  p ; 
and  if  c  be  the  modulus  of  the  tables, 

,=«"". 

If,  as  is  most  convenient,  we  conceive  the  origin  of  the  co-or- 
dinate 2,  to  be  at  the  mean  surface  of  the  earth,  or  at  the  level 
of  the  sea,  l 

dp=—sdz\  (379) 

and  substituting  the  value  of  5, 

dp= — mp  dz.  (380) 

If  P  be  the  pressure  of  the  atmosphere  at  the  level  of  the  sea, 
when  2=0,  p=P  ;  and  integrating  the  foregoing  equation,  we 
obtain 

roz=log.  P— log.  p,  (381 ) 

and 


p=Pc~     --.  (382) 

Hence : 

The  altitude  of  any  point  in  the  atmosphere,  above  the  level 
of  the  sea,  is  proportional  to  the  difference  between  the  logarithms 
of  the  respective  pressures-,  and  the  difference  of  level  between 
any  two  points  in  the  atmosphere,  is  in  a  similar  manner  propor- 
tioned to  the  difference  of  the  logarithms  of  the  pressures  at  these 
two  points.  The  column  of  mei-ury  in  the  Torricellian  tube  is 
the  measure  of  the  pressure  at  the  poiot  where  it  is  placed,  and 
therefore  the  difference  in  the  logarithms  of  the  height  of  the 
two  columns,  at  the  two  points,  is  in  like  manner  proportioned 
to  their  difference  of  level. 

In  the  equation  (380),  if  p=0,  z  becomes  infinite  ;  hence  we 
reach  the  conclusion  already  stated,  of  the  infinite  extent  of 
the  atmosphere,  when  the  relations  of  air  of  different  densities 
to  temperature  is  not  taken  into  account. 

367.  If  the  temperature  be  not  constant,  as  it  is  not  in  practice, 
it  becomes  necessary  to  take  into  view  the  dilatation  and  contrac- 
tion of  the  air  by  changes  of  temperature.  We  owe  the  original 
determination  of  the  law  of  the  dilatation  of  air  to  Gay  Lussac. 
Similar  inferences  were  obtained  separately,  at  about  the  same 


Book   V.]  PERMANENTLY  ELASTIC  FLUIDS.  365 

period,  by  Dalton.  Still  more  accurate  experiments  have  since 
been  performed  by  Petit  and  Dulong.  It  is  not  our  province  to 
enter  into  the  detail  of  this  inquiry,  but  it  is  sufficient  to  express 
the  general  law,  which  is  : 

All  permanently  elastic  fluids  expand  an  equal  proportion  of 
their  volume,  estimated  at  some  given  temperature,  for  equal  in- 
crements of  temperature,  and  this  proportion  is  the  same  in  them 
all.  Within  the  usual  limits  at  which  experiments  or  observa- 
tions are  made,  the  expansion  of  mercury  in  a  glass  tube,  or  its 
rise  in  the  thermometer,  is  also  uniform  :  hence,  every  perma- 
nently elastic  fluid  expands  an  equal  and  constant  quantity  for 
every  degree  of  thermometric  heat,  and  this  is  usually  estimated 
in  the  form  of  a  fraction  of  its  bulk  at  the  temperature  of  melting 
ice.  The  experiments  of  Petit  and  Dulong  make  this  fraction 
0.002083  for  every  degree  of  Fahrenheit's  thermometer. 

368.  The  two  laws  of  Mariotte  and  Gay  Lussac  combined, 
will  enable  us  to  determine  the  density  of  atmospheric  air,  under 
any  pressure  and  at  any  temperature,  its  density  at  some  given 
pressure  and  temperature  being  first  known. 

If  a  be  the  constant  co-efficient  of  dilatation,  0.00208  ;  i  the 
number  of  degrees  of  the  thermometer  above  or  below  32'  ;  any 
given  unit  of  bulk  of  air,  under  the  mean  pressure  of  the  atmos- 
phere, and  at  the  temperature  of  32°,  will  become  at  the  tempera- 
ture t, 


and  if  P  be  the  mean  pressure  of  the  atmosphere,  and  P'  any 
other  pressure,  this  bulk  will,  by  the  law  of  Mariotte,  become 
(±at)P 
-p7—  •  (383) 

As  the  volumes  are  inversely  as  the  densities,  if  D  be  the  den- 
sity under  the  pressure  P,  and  temperature  32°;  and  D'  the  density 
at  the  pressure  P',  and  temperature 


whence 

(384> 

And  so  in  respect  to  any  other  gas,  whose  density  at  the  mean 
pressure  P,  and  temperature  32°,  is  D",  we  obtain  for  its  density 
D'",  at  the  temperature  <,  and  pressure  P', 


366  EQUILIBRIUM  OF  [Book  V. 

and  by  comparison  with  the  last  formula,  (384), 
DP' 
j)"  —  £)'"•  (386) 

Thus  the  densities  of  gases  preserve  the  same  relation  to  each 
other  at  every  temperature  and  every  pressure. 

369.  By  reference  to  the  investigations  of  Chapter  III.,  and 
comparing  them  with  the  characters  of  elastic  fluids,  it  will  be 
seen  that  they  possess,  in  some  respects,  properties  in  common 
with  liquids.  Thus  : 

(1.)  The  pressure  of  an  elastic  fluid  will  be  proportioned  to 
the  surface  on  which  it  presses,  and  is  measured  by  the  weight 
of  a  prism  of  the  fluid,  of  uniform  density  throughout,  whose  base 
is  equal  to  the  surface  pressed,  and  whose  altitude  is  such,  that  if 
the  fluid  were  of  uniform  density,  the  pressure  on  the  unit  of 
surface  would  be  the  same  as  it  is  found  to  be  under  the  law  of 
variable  density. 

(2.)  A  solid  body,  immersed  in  an  elastic  fluid,  loses  as  much 
weight  as  is  equal  to  the  weight  of  the  fluid  it  displaces,  or  of  a 
volume  of  the  fluid  equal  to  its  own.  Hence  the  intense  pres- 
sure of  15  pounds  on  each  square  inch  of  the  surface  of  our  bo- 
dies, and  which  on  the  body  of  a  middle-sized  man,  amounts,  if 
the  mere  intensity  of  the  pressures  be  estimated,  without  refer- 
ence to  their  counteracting  directions,  to  11  tons,  has  for  its  re- 
sultant, §  336,  only  the  weight  of  a  mass  of  air  equal  to  our  bo- 
dies in  volume;  and  as  the  direction  of  this  resultant  is  upwards, 
so  far  from  being  crushed  by  this  pressure,  we  are  actually  sup- 
ported by  it.  If,  however,  the  pressure  be  removed  from  any 
part  of  our  bodies,  it  becomes  manifest  on  the  rest,  and  presses 
the  part  to  which  the  exhaustion  is  applied  against  the  apparatus 
that  operates  the  exhaustion.  Such  is  the  case  of  the  experiment, 
§  361,  No.  2. 

This  buoyancy  of  the  air  will  render  the  determination  of  the 
specific  gravity  of  bodies,  by  the  methods  of  Chap.  IV.,  inaccu- 
rate ;  for  instead  of  their  absolute  weight,  we  obtain  the  weight 
less  that  of  an  equal  bulk  of  air.  A  correction  may,  however,  he 
applied  to  the  approximate  specific  gravity,  which  rests  on  the 
following  principles. 

Let  to  be  the  absolute  weight, 

\V      the  weight  in  air, 

W       the  weight  in  water, 

S         the  specific  gravity  as  usually  obtained, 

»'         the  true  specific  gravity, 

m         the  specific  gravity  of  air. 

The  absolute  weight  will  be  greater  than  the  weight  in  air  by  the 
weight  of  a  mass  of  air  equal  in  bulk  to  the  body.      As  it  should 


Book   V.]  PERMANENTLY  ELASTIC  FLUIDS.  367 

lose  in  water  w  —  W,  it  will  lose  in  air  m  (w  —  W),  we  have  for 
the  value  of  the  absolute  weight, 

'). 


We  may,  without  sensible  error,  substitute  W  for  u>,  in  the 
quantity  w  —  W',  and  the  equation  becomes 
w>=W-t-m(W—  W'). 

We  may  then  apply  the  correction  m  (W  —  W')  to  the  weight 
in  air,  before  dividing  by  the  loss  of  weight  ;  and  the  latter  may 
be  taken  in  reference  to  this  corrected  weight  instead  of  the 
weight  in  air. 

The  value  of  m  will  be  determined  hereafter. 

The  exact  value  of  the  absolute  specific  gravity  will  be 

5=S+Wl(l—  S). 


368  EQUILIBRIUM  AND  [Book   V. 


CHAPTER  VIII. 

OF  THE  EQUILIBRIUM  AND  TENSION  OF  VAPOURS. 

370.  Water,  which  boils  under  the  mean  pressure  of  the  atmo- 
sphere at  212°,  boils  under  diminished  pressures  at  a  lower  tem- 
perature, and  may,  by  increased  pressures,  be  prevented  from 
boiling  until  it  attain  a  greater  degree  of  sensible'  heat.  The 
same  is  the  case  with  other  liquids;  each  has  a  determinate  de- 
gree of  temperature  at  which  it  boils,  under  the  mean  pressure 
of  the  atmosphere ;  and  the  boiling  point  is  raised  or  lowered, 
with  the  increase  or  diminution  of  the  pressure  to  which  it  is 
subjected. 

In  the  ebullition,  an  elastic  fluid  is  generated,  and  it  is  only 
when  the  tension  of  this  elastic  fluid  becomes  equal  to  the  pres- 
sure, that  the  action  called  boiling,  takes  place.  These  elastic 
fluids  may,  by  the  abstraction  of  their  latent  heat,  be  restored  to  the 
liquid  form,  and  hence  belong  to  the  subdivision  we  have  styled 
vapours. 

Although  that  rapid  formation  of  vapour,  which  is  attended 
with  ebullition,  takes  place  under  a  given  pressure,  at  a  con- 
stant temperature  in  each  different  liquid  ;  still  the  liquid  is  ca- 
pable of  rising  in  the  form  of  vapour  at  lower  temperatures. 
Thus,  although  water,  under  ordinary  circumstances,  does  not 
boil  below  212°,  yet  if  left  exposed  in  an  open  vessel,  a  gradual 
diminution  of  its  bulk  will  take  place  at  lower  temperatures,  and 
it  will  finally  be  wholly  dissipated.  This  has  been  conclusively 
shown  by  Dalton,  to  be  owing  to  the  formation  of  vapour  of  a 
temperature  identical  with  that  of  the  water  whence  it  proceeds; 
and  of  an  elastic  force,  identical  with  that  which  would  proceed 
from  water,  boiling  at  that  temperature,  under  diminished  pres- 
sure. Even  ice  is  capable  of  furnishing  vapour  of  its  own  tempe- 
rature, and  an  appropriate  tension.  The  same  is  the  case  with 
other  liquids;  although  all  boil  under  given  pressures,  at  some 
temperature  constant  in  each  ;  all  give  out  vapour  insensibly  at 
less  elevated  temperatures. 

371.  To  measure  the  tension  of  aqueous  vapour,  a  portion  of 
water  may  be  introduced  into  the  Torricellian  apparatus,  where  it 
will  float  upon  the  surface  of  the  mercury.  Vapour  of  the  tem- 
perature of  the  water,  will  immediately  form  in  the  vacuum.  Its 
pressure  will  be  indicated  by  its  causing  the  mercury  to  descend 
from  its  original  height ;  the  tension  of  the  vapour  will  be  mea- 


Book  V.~\  TENSION  OF  VAPOURS.  369 

sured  by  the  space  through  which  the  mercury  falls ;  or  by  the 
difference  in  the  height  of  the  mercury  in  the  tube  in  which  the 
experiment  is  performed,  and  that  at  which  it  stands  in  the  baro- 
meter. 

If  the  tube  be  of  considerable  length,  and  if,  instead  of  a  basin, 
another  tube  of  larger  diameter  than  the  former  be  used  to  hold 
the  mercury  in  which  it  is  inverted,  the  pressure  upon  the  vapour 
contained  in  the  Torricellian  vacuum,  may  be  varied.  Thus  :  By 
raising  the  inverted  tube  the  pressure  may  be  diminished  ;  in  this 
case  it  is  found  that  so  long  as  any  water  remains,  new  vapour 
of  the  same  tension  will  be  generated,  and  the  altitude  of  the  mer- 
cury in  the  tube  will  remain  constant.  If  the  tube  be  depressed, 
the  increase  of  pressure  causes  the  condensation  of  a  portion 
of  the  contained  vapour,  and  the  mercury  will  almost  instantly 
rise  to  its  original  height  above  the  surface  of  the  mercury  in 
the  outer  tube ;  or  if  the  depression  be  performed  slowly,  the 
mercury  will  move  up  the  inner  tube  exactly  in  proportion  as 
the  latter  is  pressed  down,  and  will  stand  at  a  constant  altitude. 

If  the  quantity  of  water  in  the  tube  be  so  small  that  in  raising 
the  tube  the  whole  will  be  converted  into  vapour,  it  will  be  found 
that  after  that  time  the  tension  of  the  confined  vapour  diminishes, 
as  does  that  of  air,  with  the  increase  of  the  space  it  occupies. 

It  may  hence  be  inferred  that  the  law  of  Mariotte  is  inapplica- 
ble to  aqueous  vapour,  so  long  as  it  is  in  contact  with  the  liquid 
whence  it  is  generated  ;  but  this  law  becomes  true  at  diminished 
pressures,  when  no  water  is  present.  The  same  is  found  to  be 
true  in  the  case  of  all  the  vapours,  on  which  experiment  has  been 
made.  All  vapours  then,  have,  at  a  given  temperature,  a  max- 
imum of  density  and  tension  that  cannot  be  exceeded  ;  and  on 
attaining  which,  they  are  condensed  into  the  liquid  form  by  any 
attempt  to  compress  them. 

The  essential  characteristic  of  a  vapour,  therefore,  is,  that  at  a 
given  temperature,  no  more  than  a  limited  quantity  by  weight 
can  exist  in  a  given  space  ;  so  that  by  diminishing  the  space,  the 
whole  excess  is  reduced  by  pressure  to  the  liquid  form,  without 
any  increase  taking  place  in  the  tension.  While,  when  the 
space  that  gases  occupy  is  diminished  by  pressure,  their  tension, 
at  a  constant  temperature,  is  increased  exactly  as  the  space  they 
occupy  is  lessened. 

372.  It  may  be  easily  ascertained  in  the  course  of  these  expe- 
riments, that  the  tension  of  the  vapour  is  increased  by  an  increase 
of  temperature,  and  vice  versa.  We  shall  hereafter  inquire  into 
the  law  that  this  increase  follows.  If  then  vapour  be  formed,  or 
admitted  into  a  space  unequally  heated,  it  might  at  first  sight  ap- 
pear that  the  vapour  would  have  unequal  tensions.  This,  how- 

47 


370  EQUILIBRIUM  AND  [Book    V. 

ever,  is  not  the  fact,  for  in  consequence  of  its  elasticity  and  fluid 
nature,  it  would  tend  to  an  equilibrium  of  tension  ;  the  warmer 
portions  passing  to  the  parts  of  the  space  that  are  less  heated, 
would  have  their  temperature  lowered,  and  tension  diminished  ; 
and  thus,  in  a  space  unequally  heated,  the  maximum  of  tension, 
when  equilibrium  is  established,  is  no  more  than  what  is  due  to 
the  minimum  temperature. 

373.  When  a  vapour,  after  being  formed,  is  heated  out  of  con- 
tact with  any  of  the  liquid  whence  it  is  generated,  it  tends  to  ex- 
pand, and  its  elastic  force  is  increased  exactly  as  if  it  were  a  gas. 

Hence,  at  a  constant  density,  its  pressure  becomes,  §  368, 

F=P(l=FoO;  (387) 

and  under  a  constant  pressure,  its  density  is,  §  368, 

D 

(388) 

But  when  a  vapour  is  heated  in  contact  with  the  liquid  whence 
it  is  generated,  the  latter  acquires  the  power  of  furnishing  vapour 
until  the  maximum  of  tension  due  to  that  temperature  is  attained. 
Thus  not  only  does  the  expansive  force  increase  with  the  tempe- 
rature, according  to  the  law  of  Gay  Lussac,  but  with  the  density 
caused  by  the  new  evaporation,  according  to  the  law  of  Mariotte. 

The  relation  between  the  temperatures,  densities,  and  tensions 
of  vapours  might,  therefore,  seem  susceptible  of  determination. 

In  effect,  in  the  case  of  water  which  boils  at  180°  above  the 
freezing  point,  an  investigation  analogous  to  that  of  §  367, 
gives  us, 

^      ^    F  (1  +  180  a) 

D=D    P    (1+oQ       •  (389) 

We  are  however  unable  by  this,  to  determine  the  tensions  inde- 
pendently of  the  densities,  and  are,  therefore,  compelled  to  have 
recourse  to  experiment,  in  order  to  determine  the  elastic  force  of 
steam  and  other  vapours  at  given  temperatures.  From  these  ex- 
periments empirical  formulae  may  be  deduced. 

374.  The  best  experiments  on  the  tension  of  aqueous  vapour 
at  low  temperatures,  are  those  of  Dalton,  which  are  comprised  in 
the  following  table. 


Book  V.] 


TENSION  OP  VAPOURS. 


371 


TABLE 

Aqueous  Vapour,  at  temperatures  below 
of  the  column  of  Mercury  they  are  capa- 


Te.perature. 

117° 
122° 
127° 
132° 
137° 
142° 
147° 
152° 
157° 
162° 
167° 
172° 
177° 
182° 
187° 
192° 
197° 
202° 
207° 
212° 


Of  the  Maximum  Tension  of 
212°,  estimated  in  the  height 
ble  of  supporting. 

Temperatur,  j  *•*%£ 

2°  0.068 

12°  0.096 

0.115 
0.168 
0.200 
0.237 

42°  0.283 

47°  0.339 

52°  0.401 

57°  0.474 

62°  0.560 

67"  0.655 

72°  0.770 

77°  0.910 

82°  LOT 

87°  1.24 

92°  1.44 

97°  1.68 

102°  1.98 

107°  2.32 

112°  2.68 

From  the  numbers  in  this  table  the  following  formula  for  the 
elastic  force  P,  has  been  deduced,  in  terms  of  degrees  of  the  tem- 
perature /,  reckoned  from  32°, 

P  =  .1781(l-f-.00607.  (390) 

A  formula  better  adapted  to  atmospheric  temperatures,  is 

P=.18+.007  1  +.00019  P  (391) 

The  tension  of  aqueous  vapour  or  steam,  at  temperatures  above 
the  boiling  point,  is  usually  estimated  in  atmospheres.  The  la- 
test experiments  are  those  of  Dulong  and  Arago,  which  give  the 
results  of  the  following  table,  as  far  as  24  atmospheres  from  ob- 
servation, and  as  far  as  50  from  calculation. 

The  formula  used  in  the  calculations,  and  which  is  deduced 
from  the  observations,  is 

P  =  (l  +  .  00397  O5;  (392) 

t  being  estimated  from  32°. 


3.09 

3.50 

4.00 

4.60 

5.29 

6.05 

6.87 

7.81 

8.81 

9.91 

11.25 

12.72 

14.22 

15.86 

17.80 

19.86 

22.13 

24.61 

27.20 

30.00 


372  EQUILIBRIUM  AND  [Book   V. 

TABLE 

Of  the  Elastic  force  of  Steam  at  corresponding  temperatures,  from  1  to 

50  Atmospheres. 

Tension  in     I  Corresponding  I    Tension  in     I  Corresponding 
Atmospheres.  |  Temperature.    |  Atmospheres.  |  Temperature. 

1  212°  9  350°.8 
1£  229°.2  10  358°.9 

2  250°.5  11  366°.8 
2£  263°.8  12  374°. 

3  275°.2  13  380°.6 
3i  285°.  1  14  386°.9 

4  294°  16  398°.5 
4i  301°.3  18  408°.9 

5  308°.8  20  418°.5 
54  314°.  22  427°.3 

6  320°.4  24  435°.6 
6£  326°.5  30  457°.2 

7  331°.7  40  486°.6 

8  341°.8  50  508°.6 

375.  In  experimenting  on  the  tensions  of  the  vapours  of  sub- 
stances other  than  water,  Dalton  discovered  that  a  remarkable  law 
held  good  within  the  limits  at  which  his  experiments  were  made. 
This  law  goes  by  his  name,  and  is  as  follows,  viz: 

Every  different  liquid  has  a  determinate  temperature  at  which 
it  boils  under  the  mean  pressure  of  the  atmosphere.  At  this  tem- 
perature the  elastic  force  of  its  vapour  is  just  equal  to  the  pres- 
sure of  the  atmosphere.  At  other  temperatures  equidistant  from 
the  boiling  points  of  the  different  liquids,  their  respective  tensions 
are  still  equal. 

Thus  :  water  boils  at  212°,  and  ether  at  96°,  the  tension  of  the 
vapours  being  in  both  cases  the  same  ;  when  water  is  heated  un- 
der pressure  to  250°. 5,  its  tension  is  doubled,  for  an  increase  in 
temperature  of  38°. 5 ;  and  so,  the  vapour  of  ether  at  the  tempera- 
ture of  96°-r-38°, 5=134°. 5,  has  an  elastic  force  equivalent  to 
two  atmospheres. 

376.  By  an  apparatus  similar  in  principle  to  the  tube  of  Mari- 
otte,  §365,  (2)  air  may  be  enclosed,  and  subjected  to  any  given 
pressure.     If  this  air  be  perfectly  dry,  and  if  water  be  passed  up 
into  the  space  occupied  by  the  air,  it  will  be  found  that  it  evapo- 
rates at  all  temperatures  whatsoever,  furnishing  a  vapour  whose 
tension  may  be  determined  by  the  change  it  causes  in  the  column 
of  mercury  that  confines  and  compresses  the  air.      The  rapidity 
with  which  this  vapour  is  given  out,  will  be  affected  by  the  pres- 
sure ;  being  instantaneous,  as  has  been  shown,  in  vacuo,  and  be- 
coming less  rapid  for  every  increase  in  the  pressure. 


Book  F.]  TENSION  or  VAPOURS.  373 

After  a  time,  however,  the  pressure  of  the  vapour  adds,  to  the 
original  tension  of  the  confined  air,  an  elastic  force  which  is  ex- 
actly equivalent  to  the  maximum  tension  of  vapour  of  the  tem- 
perature at  which  the  experiment  is  made.  The  evaporation  then 
ceases,  and  the  joint  tension  of  the  confined  air  and  vapour  re- 
mains constant. 

Thus  the  presence  of  air  does  not  effect  the  maximum  tension 
of  vapour  of  a  given  temperature,  but  merely  retards  its  forma- 
tion; and  the  quantity  of  aqueous  matter  in  the  elastic  form  that 
can  exist  in  a  given  space  is  the  same,  whether  that  space  be  void 
of  any  other  substance,  or  filled  with  air  of  any  density  whatsoever. 
Vapour  then,  like  a  gas,  tends  to  distribute  itself  around  the  earth 
in  an  atmosphere,  and  the  formation  of  such  an  atmosphere  is  not 
prevented,  but  merely  retarded,  by  the  presence  of  an  aeriform  at- 
mosphere. 

Vapour  when  mingled  with  a  permanently  elastic  fluid,  may 
be  condensed  by  the  same  two  causes  that  induce  its  precipitation 
in  vacuo  :  namely,  by  an  increased  pressure,  and  by  a  diminished 
temperature. 

The  laws  that  are  applicable  to  the  mixture  of  two  gases,  or  of 
a  gas  with  a  vapour,  are  true  of  the  mixture  of  any  number  of  elas- 
tic fluids,  whether  they  be  permanently  so  or  not.  And  thus  in 
a  given  space,  any  number  of  elastic  fluids  whatever  may  be  en- 
closed, without  these  substances  interfering  with  each  other.  It 
is  only  necessary  that  they  be  subjected  to  a  pressure  equal  to  the 
sum  of  their  several  tensions. 


374  SPECIFIC  GRAVITY  [Book   V. 

CHAPTER  IX. 

OF  THE  SPECIFIC  GRAVITY  OF  ELASTIC  FLUIDS. 

377.  We  may,  as  has  been  stated  in  §363,  obtain  the  weightof 
a  mass  of  atmospheric  air,  by  taking  a  flask  furnished  with  a  stop- 
cock, weighing  it,  and  then  adapting  it  to  the  plate  of  the  air- 
pump  to  exhaust  the  air.  The  stop-cock  being  closed,  the  flask 
may  be  removed,  and  again  weighed  ;  the  difference  between  its 
weight  under  the  two  different  circumstances,  is  the  weight  of  the 
air  that  has  been  withdrawn. 

As  the  air  of  the  atmosphere  always  contains  moisture,  a  more 
correct  measure  of  the  weight  of  the  air  the  flask  is  capable  of 
containing,  may  be  obtained,  by  taking  the  last  weight  as  the  ab- 
solute weight  of  the  flask;  air  that  has  been  carefully  dried  by 
hygrometric  substances  is  then  introduced,  and  the  flask  again 
weighed.  We  thus  obtain  its  weight,  free  of  the  influence  of  the 
vapour  it  contains  under  ordinary  circumstances. 

The  weight  of  any  other  gas  may  be  ascertained  in  a  similar 
manner,  by  introducing  it,  after  being  well  dried,  into  a  flask, 
whence  the  air  has  been  previously  exhausted. 

In  performing  these  experiments,  it  will  be  obvious  that  the 
quantity  of  gas  that  will  enter  the  vessel,  depends  upon  the  pres- 
sure at  which  it  is  introduced,  and  upon  its  own  temperature  ;  for 
under  different  pressures  it  will  have  different  densities,  the  tem- 
perature remaining  constant  ;  and  at  different  temperatures  the 
density  will  also  vary,  the  pressure  remaining  constant.  Hence, 
both  the  temperature  and  pressure,  at  which  the  experiments  are 
made,  must  be  noted. 

The  flask  in  which  the  experiment  is  made,  will  also  vary  in 
size,  under  the  influence  of  temperature  ;  the  effect  thus  produced, 
in  consequence  of  the  dilatability  of  the  glasss,  must  therefore  be 
taken  into  account. 

If  the  gas  cannot  be  introduced  perfectly  dry,  its  hygrometric 
state  must  be  observed. 

The  operations  of  weighing  are  performed  in  the  air  of  the  at- 
mosphere ;  hence,  the  apparent  weight  of  the  flask,  whether  full 
or  empty,  will  be  less  than  the  true  by  the  weight  of  an  equal  vol- 
ume of  air.  This  loss  of  weight  will  be  affected  by  the  pressure 
and  temperature  of  the  air,  and  by  the  quantity  of  aqueous  vapour 
it  contains. 

The  air-pump  does  not  exhaust  the  whole  of  the  air  from  the 
flask,  and  hence  the  proportion  that  remains,  which  will  be  indi- 
cated by  the  guage  of  the  pump,  must  be  noted. 


Book  P.]  or  ELASTIC  FLUIDS.  375 

Such  are  the  principles  on  which  the  determination  of  the 

weight  of  atmospheric  air,  and  different  gases  depend.  The  detail 

|  of  the  operations,  and  the  manner  of  applying  the  corrections  may 

be  seen  by  reference  to  Biot :   Traite  Complet  de  Physique, 

The  capacity  of  the  flask  being  known,  the  weight  thus  ob- 
tained may  be  compared  with  that  of  an  equal  bulk  of  water,  and 
the  specific  gravity  determined  upon  the  principles  of  Chap.  IV, 
in  terms  of  that  liquid  as  a  unit.  It  has  been  thus  found  that  the 
specific  gravity  of  atmospheric  air,  at  the  temperature  of  32°,  is 
0.001299055.  Its  density  at  any  other  temperature  and  pressure, 
may  be  obtained  by  means  of  the  formula  (384).  This  fraction  is 
the  value  of  m,  in  the  formula  for  the  absolute  specific  gravity  of 
bodies,  §  369. 

This  comparison  is  necessary,  in  order  to  connect  the  tables  of 
the  specific  gravities  of  gases  with  those  of  solid  and  liquid  bodies, 
in  the  latter  of  which  water  is  employed  as  the  unit  of  density. 

When  the  specific  gravities  of  gases  are  sought,  it  is  usual  to 
determine  them  in  terms  of  some  body  of  that  class,  taken  as  the 
unit.  Atmospheric  air,  which,  when  freed  from  moisture,  has 
an  uniform  constitution  in  all  parts  of  the  globe,  is  most  conve- 
nient for  this  purpose.  It  is,  therefore,  most  frequently  employed 
in  mere  mechanical  investigations.  But  for  many  purposes  in 
chemistry,  hydrogen  possesses  superior  advantages,  particularly 
from  the  fact  of  the  numbers  that  represent  the  den-sities  being 
•whole,  in  consequence  of  the  great  levity  of  that  gas.  Some 
chemical  writers  employ  oxygen  as  the  unit. 

We  subjoin  a  table  of  the  specific  gravities  of  some  of  the  gases, 
in  terms  of  atmospheric  air. 

TABLE 

Of  the  Specific  Gravities  of  Gases. 

Atmospheric  Air            .     '  '     •  1.0000 

HydriodicGas      .     'V     '  4.4288 

Silico-Fluoric  Gas         .          *  3.5735 

Chlorine  -  V'1'-     .         .     >       <  2.4216 

Sulphurous  Acid  Gas    .  2.1930 

Cyanogen            ~»^MT.  •»'      »i  1,8064 

Nitrous  Oxide      .     i  >.n. »;  -i>  .5269 

Carbonic  Acid  Gas     ;.vv  .5245 

Muriatic  Acid  Gas    ..;.    ;»«-;•  .2474 

Sulphuretted  Hydrogen      '.  iu  .1912 

Oxygen    >,!.-"     .         .  .1026 

Nitrogen  '".'..     .         .  0.9757 


SPECIFIC  GRAvrrr  [Book  V. 

Carbonic  Oxide  .  .  0.9569 

Ammonia  .         .  .  0.5967 

Carburetted  Hydrogen  .  0.5596 

Hydrogen  .         .  .  0.0688 

378.  The  density  of  vapour  may  be  determined  by  introducing 
i  known  weight  of  the  substance  that  yields  it,  into  a  receiver 
ntaining  mercury,  supported  by  the  pressure  of  the  atmosphere, 
d  inverting  the  receiver  in  a  vessel  also  containing  mercury, 
upon  the  principle  of  the  Torricellian  experiment.     The  outer 
vessel  is  tall  enough  to  contain  a  mass  of  some  transparent  fluid 
sufficient  depth  to  cover  the  receiver.     The  whole  apparatus 
is  then  heated  ;  and  when  the  whole  of  the  substance,  whose  va- 
pour is  under  experiment,  has  been  evaporated,  the  space  the  va- 
iour  occupies  and  its  temperature  are  noted.     We  then  have  a 
Jlk  of  the  vapour,  whose  weight  is  the  same  as  that  of  the  sub- 
stance whence  it  was  generated  ;  the  temperature  is  known,  and 
>e  pressure  can  be  determined  by  means  of  the  columns  of  mer- 
cury and  of  the  surrounding  liquid  compared  with  the  indication 
the  barometer  at  the  time.   In  this  manner,  it  is  found  that  the 
density  of  steam  at  the  temperature  of  212°  is  „'„  part  of  the 
ensity  of  water  at  32°.     The  densities  of  vapours  compared 
with  atmospheric  air  as  the  unit,  are  as  follows: 

TABLE 

Of  the  Specific  Gravities  of  the  Vapours  of  Different  Substances,  at  their 
boiling  temperatures,  under  the  mean  pressure  of  the  atmosphere.  Jit- 
motphtric  air  at  the  temperature  o/32°,  and  under  the  same  pressure, 
being  taken  as  the  unit. 

Vapour  of  Iodine         ,         .  .  8.6111 

of  Oil  of  Turpentine  .  5.0130 

of  Nitric  Acid  .  .  3.1805 

of  Sulphuret  of  Carbon  .  2.6447 

of  Sulphuric  Ether  .  2.5860 

of  Pure  Alcohol       .  .  1.6133 

of  Water         .         .  .  0.6235 

379.  The  density  of  a  vapour,  under  a  given  pressure,  and  at 
a  given  temperature  being  known,  that  under  any  other  pressure, 
and  at  its  corresponding  temperature,  can  be  calculated  by  the 
formula  (389).  In  this  manner  the  density  of  steam,  in  terms 
of  water,  and  the  relative  volumes  occupied  by  given  weights  of 
aqueous  vapour,  have  been  calculated.  The  pressures,  and  the 
temperatures,  above  the  boiling  point,  have  been  taken  from  older 
experiments  than  those  of  Arago  and  Dulong.  They  have  been 
purposely  retained  in  order  to  show  the  view  of  this  subject  that 
is  still  most  generally  received. 


Book  F.] 


ELASTIC  FLUIDS. 


377 


TABLE 

Of  the  Density  and  Volume  of  Aqueous  Vapour,  at  its  Maximum  Ten- 
sion. The  Unit  of  Density  being  water  at  the  temperature  o/32°,  and 
that  of  volume,  the  volume  oj  an  equal  weight  of  water,  also  at  32°. 


32° 

41° 
50° 
59° 

68° 
770 

86° 
95° 
104° 
113° 
122° 
131° 
140° 
149° 
158° 
167° 
176° 
185° 
194° 
203° 
512° 
251.6° 
291.2° 
330.8° 
370.4° 


a 
it 

ti 
U 
t( 
it 

U 
U 

1 

2 

4 

8 

16 


0,0000053 

0,0000073 

0,0000097 

0.0000131 

0.0000173 

0.0000227 

0.0000297 

0.0000390 

0.0000499 

0.0000637 

0.0000710 

0.0001022 

0.0001261 

0.0001592 

0.0001964 

0.0002388 

0.0002936 

0.0003557 

0.0004261 

0.0005074 

0.0005900 

0.00110 

0.00210 

0.00399 

0.00760 


188600 
137000 
103000 
76330 
57800 
44050 
33670 
25640 
20030 
15690 
14080 
9784 
7930 
6281 
5091 
4187 
'  3406 
2811 
2346 
1971 
1696 
909 
476 
250 
131 


traeted  from  that  of  Darnell. 


48 


SPECIFIC  GRAVITY,  &C.  [Book     V. 

TABLE 

Of  the  Force,  Weight,  and  Expansion  of  Aqueous  Vapour,  at  different 
degrees  of  temperature,  from  0°  to  92°. 


Temperature. 

Elastic  force  in 
inches  of  mercury. 

Weig-ht  of  a  cubic  1  ., 
foot  in  grains.     |  Evasion. 

0° 

0.064 

0.789 

.000 

12° 

0.096 

1.156 

.024 

22° 

0.139 

1.642 

.045 

32° 

0.200 

2.317 

.066 

42° 

0.283 

3.214 

.087 

52° 

0.401 

4.468 

.108 

62° 

0.560 

6.126 

.129 

72° 

0.770 

8.270 

.150 

82° 

1.070 

11.293 

.170 

92° 

1.440 

14.931 

.191 

212° 

30.000 

257.191                1.441 

380.  The  presence  of  aqueous  vapour  in  atmospheric  air,  may  be 
shown  by  gradually  cooling  down  a  polished  surface.  When  such 
a  surface  reaches  the  temperature  that  corresponds  to  the  max- 
imum tension  of  the  vapour,  a  thin  film  of  dew  will  be  deposited, 
and  cloud  the  surface.  If  the  temperature  of  the  surface  be 
ascertained,  we  can  by  means  of  the  above  table  determine  the 
quantity  of  vapour  contained  in  each  cubic  foot.  Were  the  ta- 
ble complete  to  every  degree  of  the  thermometer,  simple  in- 
spection would  give  us  the  weight  in  grs.  opposite  to  the  observed 
temperature  of  the  surface,  provided  the  air  were  also  of  that 
temperature.  But  as  the  temperature  of  the  air  is  always  higher, 
the  vapour  will  be  expanded,  and  hence,  the  weight  given  in  the 
third  column,  requires  to  be  corrected,  by  multiplying  it  by  the 
fractions,  representing  the  ratio  of  the  two  expansions,  at  the 
temperatures  of  precipitation,  and  of  the  air. 

The  temperature  of  precipitation,  or  that  to  which  the  surface 
is  reduced  at  the  moment  it  begins  to  be  clouded,  is  called  the 
Dew-Point. 

The  best  instrument  that  has  yet  been  contrived  for  observing 
it,  is  the  Hygrometer  of  Daniell. 

Another  instrument,  also  well  fitted  for  the  purpose,  has  more 
recently  been  invented  by  Pouillet.  It  is  foreign  to  our  pur- 
pose to  enter  into  the  detail  of  the  structure  of  these  beautiful 
and  ingenious  instruments. 


Book    V.]  BAROMETER  AND  ITS  APPLICATIONS.  319 


CHAPTER  X. 

OF  THE  BAROMETER  AND  ITS  APPLICATIONS. 

381.  The  Barometer  is  constructed  by  giving  to  the  Torricel- 
lian apparatus  a  support  that  unites  its  tube  and  basin  ;  its  form 
may  also  be  changed  in  such  a  manner  as  to  substitute  a  more 
convenient  receptacle  for  the  mercury  than  the  latter.  A  scale 
is  then  adapted,  by  means  of  which  the  altitude  of  the  column  of 
mercury  may  be  measured  in  some  conventional  unit  of  length. 

The  French  use  in  their  barometers,  the  metre  as  the  unit,  and 
the  scale  exhibits  its  decimal  Divisions.  The  English  divide  their 
scale  into  inches,  and  these  are  subdivided  decimally.  The  former 
assume  for  the  mean  height  of  the  mercury  in  the  barometer,  at 
the  level  of  the  sea,  0.760  metres  ;  the  latter,  30  English  inches. 
These  heights,  however,  even  when  the  pressure  remains  inva- 
riable, are  affected  by  the  change  that  takes  place  in  the  density 
of  the  mercury,  under  changes  of  temperature. 

When  the  barometer  is  intended  to  remain  stationary,  in  a 
single  place,  the  original  form  of  a  wide  basin,  in  which  the  end 
of  the  tube  is  immersed,  is  still  used  ;  and  as  a  considerable  change 
in  the  height  of  the  mercury  in  the  tube  will  not  produce  any 
sensible  difference  of  level  in  the  basin,  it  has  the  advantage  of 
needing  no  correction. 

If  the  open  end  of  the  tube  be  bent  upwards,  it  is  called  the 
syphon  barometer,  and  the  pressure  of  the  air  upon  the  mercury 
in  the  open  branch  of  the  tube,  will  produce  the  same  effect  as  it 
does  when  acting  upon  that  in  the  basin,  in  the  original  apparatus 
of  Torricelli.  A  scale  of  the  same  measure  of  length  must  be 
adapted  to  each  branch  of  the  tube,  and  the  position  of  the  mer- 
cury noted  upon  both,  in  order  to  determine  the  difference  of 
level. 

To  increase  the  length  of  the  divisions  that  correspond  to  a 
given  change  of  level  in  the  mercury,  various  plans  have  been 
proposed  ;  all,  however,  except  two,  have  gone  wholly  out  of 
use,  and  therefore  require  no  description.  These  two  are  the 
wheel  barometer  of  Hooke,  and  the  conical  barometer. 

The  form  of  the  wheel  barometer,  is  as  follows  :  adapt  a  float 
of  iron  to  the  open  branch  of  the  syphon  barometer,  and  counter- 
poise it  by  a  weight  attached  to  a  cord  passing  over  a  pulley ; 
the  weight  must  be  of  such  magnitude  that  when  the  mercury 
subsides  in  the  tube,  the  iron  float  shall  preponderate  and  follow 


380  BAROMETER  AND  \Eook    V. 

the  mercury  in  its  descent  ;  but  when  the  mercury  rises,  the  float 
being  buoyant  upon  it,  is  drawn  up  by  the  counterpoise.  In  this 
motion  the  pulley  will  be  turned  around,  and  if  an  index  be  affixed 
to  its  axis,  the  latter  will  traverse  around  a  dial  concentric  with 
the  pulley.  On  this  dial  the  divisions  may  be  marked. 

The  conical  barometer  is  a  slender  tube  with  a  conical  bore, 
the  open  end  having  the  largest  diameter.  It  is  found  that  a 
column  of  mercury  will  remain  suspended,  in  such  a  tube,  by  the 
pressure  of  the  atmosphere,  if  it  be  carefully  inverted,  and  be  not 
agitated.  This  column  will  assume  a  length  which  corresponds 
to  the  pressure  of  the  atmosphere.  If  this  pressure  be  lessened, 
the  mercury  will  fall  until  the  column  of  mercury,  which,  as 
it  descends  into  a  wider  part  of  the  tube,  must  decrease  in  alti- 
tude, is  again  in  equilibrio  with  atmospheric  pressure. 

If  the  pressure,  on  the  other  hand,  increase,  the  mercury  will 
be  forced  up;  but  the  length  of  its  column  will  increase,  in  con- 
sequence of  its  entering  a  portion  of  the  tube  of  less  diameter, 
and  the  rise  will  cease  when  this  length  becomes  the  measure  of 
the  increased  pressure. 

It  will  be  obvious  that  in  both  these  cases,  the  change  in  the 
position  of  the  mercury  must  be  considerably  greater  than  that 
which  will  take  place  in  a  tube  of  uniform  bore. 

3S2.  The  invention  of  the  Vernier  has,  in  a  great  degree,  re- 
moved the  necessity  of  seeking  for  a  form  of  the  barometer  that 
shall  have  a  scale  of  greater  length  than  that  which  corresponds 
to  the  change  of  level  in  a  tube  of  uniform  bore,  placed  in  a  ver- 
tical position.  The  scale  of  the  barometer  being  fixed,  the  ver- 
nier consists  in  a  moveable  scale  that  slides  along  it,  and  whose 
lower  or  upper  extremity  carries  the  index  that  is  made  to  cor- 
respond with  the  surface  of  the  mercury  in  the  tube.  The  length 
of  this  sliding  scale  is  made  exactly  equal  to  some  given  number 
of  divisions  upon  the  fixed  scale  ;  and  it  is  subdivided  into  a 
number  of  equal  parts,  exceeding,  or  falling  short,  by  one,  the 
number  of  divisions  upon  the  corresponding  length  of  the  scale. 

The  theory  of  this  instrument  is  as  follows  : 

Let  a  be  the  length  of  the  fixed  scale,  which  the  length  as- 
sumed for  the  vernier  exceeds  or  falls  short  of  by  1  division  ; 
let  n  be  the  number  of  divisions  in  a,  and  which  is  the  same  with 
the  number  of  divisions  in  the  vernier. 

The  length  of  the  vernier  will  be, 


a 
The  length  of  each  division  on  the  fixed  scale  is  -  ; 


Book   F.]  ITS  APPLICATIONS.  381 

The  length  of  each  division  of  the  vernier  will  be 
1  a      a      a 


And  as  the  length  of  each  division  of  the  fixed  scale  is  -,  the  dif- 

a 
ference  in  the  length  of  the  respective  divisions  is  ^f  -^  . 

To  take  the  case  of  the  common  barometer  ;  the  inch  is  di- 
vided into  ten  parts,  and  eleven  of  these  are  taken  for  the  length 
of  the  vernier,  which  is  divided  into  ten  equal  parts  ;  a  then  is 
equal  to  1  inch,  w=10,  and 

«_   1     .     , 
—  2-10om 

which  will  have  the  negative  sign.  Hence  the  changes  in  its  po- 
sition will  be  indicated  by  looking  down  its  scale,  and  counting 
the  number  of  that  division,  reckoned  from  the  upper  end  of  the 
vernier,  that  corresponds  with  a  division  of  the  fixed  scale.  The 
index  is  therefore  placed  at  the  top  of  the  vernier  ;  the  inch  and 
tenth  next  below  the  index,  give  the  height  to  the  first  place  of 
decimals,  and  the  second  place  is  given  by  the  indication  of  the 
vernier. 

In  barometers  where  a  greater  degree  of  accuracy  is  required, 
the  inch  is  divided  into  20  equal  parts  ;  the  vernier  is  made  equal 
in  length  to  24  of  these,  and  is  divided  into  25  equal  parts.  In 
this  case, 

a=1.25  inch, 

n=    25 
a       1.25 


The  difference  in  the  length  of  the  respective  divisions  then  is 
_i_  part  of  an  inch,  and  its  sign  is  positive  ;  hence,  the  index  is 
placed  at  the  bottom  of  the  vernier,  and  its  indications  sought  by 
counting  upwards  from  the  index,  until  that  division  be  reached, 
which  exactly  corresponds  to  a  division  of  the  fixed  scale. 

383.  In  some  of  the  applications  of  the  barometer,  it  is  neces- 
sary that  it  should  be  safely  portable  from  place  to  place.  This 
may  be  effected  in  various  ways. 

(  1  .  )  The  mercury  may  be  enclosed  in  a  leathern  bag,  adapted,  in- 
stead of  a  basin,  to  the  bottom  of  the  tube.  If  a  screw  be  applied 
beneath  the  bag,  the  mercury  may,  by  its  pressure,  be  forced  up 
until  it  strike  against  the  top  of  the  tube  ;  the  instrument  may 
then  be  carried,  in  an  inverted,  or  in  a  horizontal  position,  with- 
out risk  or  danger  from  the  striking  of  the  mercury  against  the 
top  of  the  tube. 

(2.)  The  open  end  of  a  syphon  barometer,  may  be  made  of 
two  pieces,  united  by  a  stopcock  of  a  material  that  is  not  acted 


382  BAROMETER  AND  [Book    V. 

upon  by  mercury.  On  inclining  the  tube,  the  mercury  strikes 
against  its  top  and  fills  it.  In  this  position,  if  the  stopcock  be 
closed,  the  mercury  will  not  be  able  to  escape  when  the  vertical 
position  of  the  instrument  is  restored  ;  neither  can  it  oscillate,  for 
it  will  completely  occupy  the  whole  of  the  tube. 

(3.)  In  the  barometer  of  Gay  Lussac,  the  necessity  of  a  stop- 
cock on  the  open  branch  of  the  syphon,  is  obviated  by  contract- 
ing the  tube  at  the  bend  to  very  small  dimensions,  and  continu- 
ing this  contraction  for  some  distance  up  the  closed  branch  of  the 
tube.  The  external  air  cannot  enter  through  the  tube  thus  con- 
tracted, and  it  may,  therefore,  be  safely  inverted,  and  carried 
from  place  to  place. 

(4.)  In  the  very  perfect  and  complete  portable  barometer  of 
Sir  George  Shuckburgh,  the  mercury  is  enclosed  in  a  wooden 
cistern,  closed  at  the  bottom  by  a  flexible  diaphragm  of  leather. 
This  is  moved  by  a  screw  until  the  mercury  fills  the  tube.  Over 
the  mercury  an  ivory  float  is  placed,  th*t  is  brought  by  the  action 
of  the  screw  to  a  mark  on  the  stem,  that  shows  when  the  mercury 
is  at  the  level  whence  the  divisions  on  the  scale  have  been  mea- 
sured. This  float  has  a  ring  between  it  and  the  mercury  that  is 
pressed  up  when  the  mercury  rises,  and  closes  the  opening 
through  which  the  float  passes. 

(5.)  In  the  portable  barometer  of  Englefield,  a  cistern  of  wood 
is  cemented  to  the  glass  tube,  and  the  whole  is  packed  in  a  rod 
of  wood,  of  the  size  of  a  common  walking  cane.  A  piston  works 
in  the  cistern,  by  means  of  a  screw,  and  can  be  raised  until  it 
force  the  mercury  to  the  top  of  the  tube.  The  construction  of 
this  instrument  has  been  much  improved  by  Daniell  ;  partly  in 
some  particulars  that  will  be  stated  in  the  next  section,  and  partly 
by  applying  a  cistern  of  cast-iron,  and  adapting  a  table  to  the 
instrument,  by  which  the  correction  for  the  expansion  of  the  glass 
and  mercury,  by  heat,  can  be  obtained  by  inspection.  The  com- 
parative diameters  of  the  tube  and  cistern  are  also  given,  and  the 
height  at  which  the  scale  has  been  adapted  to  the  tube.  At  all 
other  heights,  a  correction  must  be  applied  for  the  change  of  level 
in  the  cistern,  for  which  this  relation  between  these  diameters  is 
the  element. 

384.  In  filling  the  barometer  with  mercury,  and  applying  the 
scale,  a  variety  of  precautions  are  necessary.  The  mercury  must 
be  purified  by  chemical  means  from  all  extrinsic  substances,  for 
they  will  alter  its  density  ;  but  when  properly  purified,  the  den- 
sity and  character  of  the  mercury  are  always  identical. 

The  mercury  must  be  completely  purged  of  air,  and  air  must 
be  carefully  excluded  from  the  tube  ;  for  even  a  small  quantity 
of  air,  rising  to  the  top  of  the  tube,  will  produce  a  considerable 


Book   P.]  ITS  APPLICATIONS.  383 

depression  in  the  column  of  mercury.  To  separate  both  the  air 
that  is  contained  in  the  mass  of  mercury,  and  that  which  it  can- 
not fail  to  imbibe  in  the  act  of  being  poured  into  the  tube,  the 
mercury  must  be  boiled  in  the  tube  itself.  This  is  done  by  fill- 
ing the  tube  at  first  only  to  a  third  part  of  jts  length,  and  boiling ; 
mercury,  that  has  been  heated,  is  then  added  in  several  distinct 
portions,  and  each  successive  portion  is  heated  until  it  boils. 
After  the  tube  is  nearly  filled,  as  it  would  endanger  it  to  complete 
the  boiling,  the  residue  is  added  from  a  parcel  of  mercury  that 
has  been  boiled  separately.  The  reason  of  adding  mercury  that 
has  been  previously  heated,  is  to  prevent  the  glass  from  breaking 
by  being  suddenly  cooled.  The  effectual  exclusion  of  the  air 
may  be  ascertained  after  the  tube  has  been  inverted  in  the  cistern, 
by  inclining  the  tube  until  the  mercury  rises  to  the  top ;  if  it 
strike  hard,  and  with  a  sharp  sound,  the  air  has  been  completely 
driven  off. 

In  affixing  the  scale,  it  is  usual  to  set  it  by  comparison  with 
another  barometer;  but  Daniell  has,  in  all  the  barometers  made 
under  his  direction,  applied  scales  divided  by  actual  measure- 
ment from  the  surface  of  the  mercury  in  the  basin  ;  and  the  po- 
sition of  the  mercury  at  the  time  that  this  division  is  performed, 
is  noted  upon  the  outside  of  the  case  of  the  instrument,  as  the 
neutral  point,  as  has  already  been  stated  in  the  preceding  section. 

The  introduction  of  air,  when  the  barometer  is  inverted  for 
carriage,  and  again  restored  to  its  proper  position  for  use,  is  diffi- 
cult to  be  avoided,  in  barometers  of  the  usual  construction  ;  for 
there  is  no  adhesion  between  glass  and  mercury,  by  which  the 
passage  of  an  extrinsic  substance  can  be  prevented.  To  obviate 
this  defect,  Daniell  has  welded  a  ring  of  platinum  to  the  bottom 
of  the  tube  ;  between  this  and  the  mercury  there  is  an  attraction, 
that  will  prevent  the  entrance  of  air. 

Besides  the  correction  for  temperature,  a  correction  is  required 
for  a  depression  caused  in  the  mercury  by  capillary  action. 

385.  Although  the  mean  height  of  the  barometer  at  the  level 
of  the  sea  is  about  30  inches,  yet  it  is  far  from  standing  constantly 
at  that  height.  It  is  found  on  the  contrary  to  vary  in  a  greater 
or  less  degree  at  every  point  on  the  surface  of  the  earth,  and  va- 
riations of  the  same  character  are  found  to  take  place  at  all  alti- 
tudes that  have  been  reached.  These  variations  are  either  peri- 
odical or  accidental.  In  equatorial  regions  the  former  are  the 
more  important ;  but  in  temperate  climates,  the  periodic  variations 
appear,  on  a  first  inspection,  to  be  completely  masked  by  those 
which  are  accidental.  The  whole  amount  of  variation  too,  appears 
to  increase  as  we  recede  from  the  poles  to  the  equator,  although  it 
is  influenced  in  a  very  great  degree  by  local  circumstances.  Thus 


384  BAROMETER  AND 

at  New- York  the  variation  does  not  much  exceed   1J  inches, 
while  in  Great  Britain  it  is  as  great  as  3  inches. 

To  separate  the  accidental  from  the  periodic  variations  in  the 
barometer,  a  long  series  of  observations  must  be  made,  at  hours 
of  the  day  chosen  for  their  adaptation  to  the  purpose.  If  the 
mean  height  be  alone  sought,  the  hour  of  noon  is  well  suited,  and 
a  series  of  observations,  continued  for  some  years,  will  show 
whether  there  be  any  change  due  to  the  season  of  the  year.  For 
the  horary  oscillations,  more  frequent  observations  must  be  made. 
Intemperate  climates,  Ramond  has  proposed  as  best  .adapted 
to  the  purpose,  the  hours  of  9  A.  M.,  noon,  3,  and  9  P.  M. 
Daniell  has  directed  the  use  of  the  hours  3,  and  9  A.  M.,  and  3 
and  9  P.  M.  If  no  more  than  three  observations  are  to  be  made, 
the  latter  author  has  chosen  8  A.  M.,  4  P.  M.,  and  midnight. 
Under  the  equator,  Humboldt  has  shown  that  the  maximum  of 
height  takes  place  at  the  hours  of  9  A.  M.,  and  11  P.  M.,  the 
minimum  at  4  A.  M.,  and  4  P.  M.  In  Paris,  Ramond  has  shown 
that  the  times  of  maximum  and  minimum,  vary  with  the  season  ; 
in  winter  the  hours  of  the  maximum  are  9  A.  M.,  and  9  P.  M., 
of  the  minimum  at  3  A.  M  ;  in  summer  the  maxima  occur  at  8 
A.  M.,  and  IIP.  M.,  the  minimum  at  4  P.  M.  These  horary 
variations  appear  to  be  less  in  high  latitudes  than  at  the  equator, 
while  the  accidental  variations  follow,  as  has  been  stated,  a  dif- 
ferent law. 

386.  These  variations  are  the  consequence  and  indications  of 
changes  in  the  pressure  of  the  atmosphere.  Those  which  we 
have  styled  accidental,  are  from  longexperience  found  to  produce 
changes  in  the  state  of  the  weather.  The  nature  of  the  change 
portended,  is  however  different  in  different  countries,  and  there 
is  but  one  general  rule,  namely,  that  a  sudden  fall  of  the  mer- 
cury always  portends  a  high  wind. 

The  barometer  may  therefore  be  used  to  prognosticate  the  state 
of  the  weather,  and  will  be  effectual  for  this  purpose,  wherever  a 
number  of  observations  has  been  made  sufficient  to  detect  the  law, 
that  the  variations  of  the  one  follow  in  respect  to  the  other.  It 
is  also  used  to  great  advantage  on  ship-board,  to  foretell  gales  of 
wind. 

387.  As  the  barometer  furnishes  a  measure  of  the  pressure  of 
the  atmosphere,  and  as  this  pressure  varies,  §  366,  with  a  change 
in  the  distance  from  the  mean  surface  of  the  earth,  or  with  the 
altitude  of  the  place  of  observation  above  the  level  of  the  sea,  the 
barometer  is  used  for  the  purpose  of  measuring  differences  in  al- 
titude, and  ascertaining  the  absolute  height  of  places  above  the 
ocean.  The  principle  on  which  this  may  be  performed,  has  al- 
ready been  stated  in  §  366. 


Book  V.\  IT£  APPLICATIONS.  385 

If  the  atmosphere  were  of  uniform  temperature  throughout,  we 
have  from  (381),  for  the  value  of  the  difference  of  level  z, 

z=m.(log.  P — logp)  ; 

or,  as  the  columns  of  mercury  in  the  barometer  are  the  measures  of 
the  pressures,  P  and  p,  we  may  consider  those  letters  as  repre- 
senting the  number  of  inches  and  decimals  in  those  columns  re- 
spectively. 

The  value  of  the  constant  co-efficient  m,  is  determined  by  expe- 
riment, and  is,  by  the  observations  of  Gen.  Roy,  at  the  tempera- 
ture of  melting  ice,  and  estimated  in  English  measure,  10000  fa- 
thoms. Hence, 

z=  10000  (log.  P— log.p), 
or  in  feet, 

2=60000  (log.  P— log.  p)  .  (393) 

The  height  of  the  columns  of  mercury  i?  affected  by  tempera- 
ture ;  hence,  a  correction  must  be  Applied  to  the  observed 
columns  of  mercury  to  reduce  them  to  a  common  temperature. 

Mercury  expands  itself  0.0001016  of  its  bulk  at  32°,  for  every 
degree  of  heat :  hence,  if  the  temperature  of  the  columns  of  mer- 
cury be  known,  the  reduction  of  each  to  that  standard  temperature 
is  easy,  and  the  mode  obvious.  IVo  sensible  error  can  arise, 
however,  in  ordinary  atmospheric  temperatures,  from  considering 
the  above  fraction  as  the  rate  of  expansion  between  the  tempera- 
tures which  the  mercury  in  the  barometer  has  at  the  two  stations. 
Hence,  the  correction  may  be  applied  to  but  one  of  the  columns, 
and  it  is  most  convenient  to  do  so,  to  that  which  has  the  lowest 
temperature,  and  which  will  most  commonly  be  that  observed 
at  the  highest  station.  If  T/  and  t ',  be  the  two  temperatures,  the 
co-efficient  denoting  this  correction  will  be 

1  +0.0001016  (T'— t'}  .  (394) 

Difference  of  temperature  will  also  affect  the  density  of  the  air ; 
and  hence  the  difference  of  the  height  of  the  mercurial  columns 
will  not  be  the  same  at  other  temperatures,  at  each  or  either  of  the 
places,  as  it  would  be,  had  both  the  temperature  of  32°.  A  cor- 
rection is  therefore  needed  for  this  cause,  which  will  affect  the  co- 
efficient m.  Air  dilates,  as  has  been  stated,  §  367,  0.002083  of 
its  bulk,  for  every  degree  of  temperature  reckoned  from  32°.  This 
fraction,  therefore,  is  not  constant  at  all  temperatures.  It  is, 
however,  usual  to  consider  it  as  such,  and  to  apply  it,  by  means  of 
the  mean  temperature  of  the  air  at  the  two  stations,  above  the  free- 
zing point :  hence,  the  correction  consists  in  multiplying  m  by 

.002083 


or  which  is  the  same,  by 

1  +  0.0010415  (T-H— 64°)  .  (395) 

49 


386  BAIIOMETBR  AND  [Book   V. 

The  formula  (393)  therefore  becomes,  when  these  corrections 
are  taken  into  account, 

z= 10000  [1 +0. 001041  (T-H— 64)]  , 

j* P  (396) 

p  [1+0.0001016  (T— f )]  . 

This  formula  is  sufficiently  near  the  truth  for  most  of  the  cases 
that  can  occur  in  practice.  When  much  accuracy  is  required,  and 
especially  where  the  difference  of  level  is  great,  there  are  other 
circumstances  to  be  taken  into  account. 

(1).  The  mean  height  of  the  mercury  in  the  barometer  is  af- 
fected by  the  difference  in  the  apparent  force  of  gravity  at  dif- 
ferent latitudes,  according  to  the  law  in  §  100.  A  correction, 
therefore,  may  be  needed  on  this  account ;  the  element  of  which 
is  the  latitude  of  the  place.  This,  however,  is  at  most  small, 
and  is  generally  neglected. 

(2).  A  more  important  cause  of  error  exists  in  the  mixture  of 
atmospheric  air  with  aqueous  vapour,  which  is  subject  to  different 
laws  of  pressure  and  equilibrium.  It  has  been  considered  by  an 
authority  of  no  less  weight  than  that  of  Laplace,  to  be  impracti- 
cable in  the  present  state  of  our  knowledge,  to  apply  a  correction 
for  tins  circumstance.  This  difficulty  has,  however,  been  over- 
come by  Daniell,  who  has  shown  that  his  hygrometer  can  be  ap- 
plied to  the  purpose  ;  and  has  published  tables  that  accompany  it, 
by  means  of  which  this  correction  can  be  applied.  These  tables, 
and  the  principles  on  which  they  are  founded,  may  be  seen  on  re- 
ference to  the  "Quarterly  Journal  of  Science,  edited  at  the 
British  Institution",  No  25. 

3SS.  In  applying  the  barometer  as  a  measure  of  differences  of 
level,  several  precautions  are  necessary.  In  order  to  prevent  the 
observations  being  affected  by  the  periodic,  and  still  more  by 
the  accidental  variations  in  the  mean  pressure  of  the  atmosphere, 
they  should  be  made  simultaneously  at  the  two  places  whose  dif- 
ference of  level  is  sought.  Hence,  two  observers,  and  two  instru- 
ments, are  necessary. 

The  barometers  should  each,  have  a  thermometer  inclosed  in 
their  case,  in  order  to  mark  the  temperature  of  the  mercury  they 
contain.  These  are  called  the  Attached  Thermometers. 

Each  observer  should  be  furnished  with  a  separate  thermome- 
ter, to  note  the  temperature  of  the  air. 

One  of  the  observers  remaining  stationary,  the  other  may  move 
with  his  instruments  from  place  to  place,  and  thus  observe  at  a 
number  of  stations.  In  order  to  a  comparison  of  observations, 
the  observer  who  remains  at  the  same  place  should  take  observa- 
tions at  prescribed  intervals  of  time,  say  from  10'  to  15' ;  the  one 
who  proceeds  from  place  to  place,  should  note  the  time  of  each 


"•: 

Book  V.~\  ITS  APPLICATIONS.  367 

of  his  observations.  Those  nearest  in  point  of  time,  must  of  course 
betaken  as  the  objects  of  comparison. 

The  calculation  is  performed  as  follows  :  The  column  of  mer- 
cury at  the  station  whose  temperature  is  lowest,  is  to  be  corrected 
for  that  element.  This  correction  consists  in  adding  to  the 
height  of  that  column,  its  continual  product  by  the  constant  frac- 
tion 0.0001016,  and  by  the  difference  of  the  indications  of  the  two 
attached  thermometers. 

The  difference  between  the  logarithms  of  the  other  column  of 
mercury,  and  this  corrected  column  is  then  to  be  taken. 

This  difference  multiplied  by  10000,  is  the  approximate  differ- 
ence of  altitude  between  the  two  stations  in  fathoms,  and  is  ob- 
tained at  once,  by  moving  the  decimal  point  four  places  to  the 
right. 

The  approximate  difference  of  level  is  lastly  to  be  corrected, 
for  the  difference  of  the  temperature  of  the  air  at  the  two  stations 
from  32°.  This  correction  is  obtained  by  multiplying  the  excess 
of  the  mean  of  the  two  detached  thermometers  above  32°,  or  their 
defect  below  32°,  by  the  constant  fraction  0.002083,  and  the  pro- 
duct by  the  approximate  altitude.  This  correction  is  added  when 
this  mean  temperature  exceeds  32°,  and  subtracted  when  it 
is  less. 

389.  If  the  correction  for  the  moisture  of  -the  atmosphere, 
whose  element  is  obtained  from  the  indications  of  the  hygrometeV 
of  Daniell,  is  not  employed,  it  will  be  better  to  substitute  for 
the  constant  fraction  0.002083,  which  represents  the  expansion 
of  dry  atmospheric  air  for  each  degree  of  Fahrenheit's  thermome- 
ter, the  fraction  0.00244,  which  by  the  experiments  of  Gen.  Roy, 
is  consistent  with  a  mean  state  of  moisture. 

The  co-efficient  m,  which  we  have  stated  at  10000  fathoms, 
has  been  inferred  by  Raymond,  from  a  great  number  of  observa- 
tions, to  be  18336  metres,  at  the  latitude  of  45°.  This  is  equivalent 
to  10025  fathoms,  and  makes  the  number  given  in  our  formula 
in  error,  about  ?^th  part.  The  whole  change  in  the  intensity  of 
gravity  from  the  pole  to  the  equator  is,  as  has  been  shown,  §  100, 
aigth  part  of  the  force  of  gravity,  on  the  hypothesis  of  the  earth's 
having  a  spherical  figure  ;  but  in  consequence  of  the  spheroidal 
figure  of  the  earth,  the  apparent  intensity  is  still  farther  lessened 
at  the  equator,  and  the  ratio  of  the  two  forces  is,  §  290,  actually 
as  great  as  T|¥,  or  more  exactly  0.005674. 

As  the  value  of  m  is  determined  for  the  latitude  of  45°,  it  be- 
comes necessary  to  reduce  observations  at  otherlatitudes,  to  the  lati- 
tude of  45°.  The  value  of  the  correction  may  be  thus  found  : 

Let  g  be  the  force  of  gravity  at  the  equator;  /the  centrifugal 
force  there,  will  be  =0.005674  g. 


388  BAROMETER  AND  [Book    V. 

From  (294),  we  have  for  the  value  of  y,  the  force  of  gravity  in 
lat.  45°, 

7 
whence 

for  the  force  g',  at  any  other  latitude  L,  we  have 

g-'=g+/sin.a  L  ; 
substituting  the  value  of  g,  from  the  foregoing  equation, 

or 

but  as 

cos.  2  L=  1—2  sin.3  L; 
we  obtain 


and  substituting  the  value  of/, 

g'=y-_0.002837  cos.  2  L  . 

Whence  a  correction  may  be  deduced  to  be  applied  to  the  quan- 
tity z,  in  the  formula,  when  the  height  is  considerable,  and  the 
latitude  distant  from  45°. 

The  formula  for  the  complete  calculation  then  becomes, 
2=10000  (1—0.002837  cos.  2  L)  . 


r  /  \  H 

I  1+0.002083  (-2~  —  32)  J  . 


p 

log* 


KI+O.OOOIOGI  (T—  0]    ' 

We  annex  a  form  of  the  calculation 

EXAMPLE. 

Calculation  of  the  difference  in  level  of  two  stations,  A  and  B,  at  which 
the  following  observations  were  made  cotemporaneously. 


A  28.691  T=84°  T'=76°.5 

B  28.791  *=78°  l'=82° 


=49°    T'— f=— 5°.5     2L=91°.32' 


Book  F.]  ITS  APPLICATIONS.  389 

log.  0.  002837=  7.45286  log.  28.691  =  1.45775 
log.  cos.  910.32'=— 8.42746  log.  27.791  =  1.44390 
log.— 0.000076=  5.88032 

log.  [H-(0.0001061  X— 5.5)=0.99946]=       9,99975 

1.44365 
141  =  10000X0.01410 

log. 141=2.14922 

log.  [1  +  (0.002083X49)  =  1.10207]=0.04221 

log.  [(1—0.000076) =0.999924]  =9.99999 

log.      -  -         -         155.39  fathoms  =2.19142 

6 


932.34  feet. 

If  we  take  m=60l50  feet,  the  last  part  of  the  calculation  will 
be, 

log.  0.0140     =8.14922 

log.    60150  =4.77924 

log.  1.10207  =0.04221 

log.  0.999924=9.99999 

log.        934.68ft.      =2.97066 


39°  ATTRACTION  OP  [Book  !>r. 

CHAPTER  XI. 

OF  THE  ATTRACTION  OF  COHESION. 

390.  The  conditions  of  the  equilibrium  of  fluids  that  have  been 
investigated  in  the  preceding  chapters,  are  occasionally  affected 
by  an  action  that  takes  place  between  their  particles,  and  those  of 
solid  bodies  in  contact  with  them.     This  action  is  called  the  At- 
traction of  Cohesion,  or  from  the  most  remarkable  class  of  the 
phenomena  to  which  it  gives  rise,  Capillary  Attraction. 

The  existence  of  this  attraction,  and  its  capability  of  exerting 
a  determinate  force,  may  be  shown  by  a  very  simple  experiment. 

If  a  disk  of  any  substance  that  is  capable  of  being  moistened  by 
a  liquid,  be  suspended  from  the  arm  of  a  balance,  and  counter- 
poised by  weights  ;  and  if  a  vessel  containing  the  liquid  be  raised 
from  beneath,  until  its  surface  just  touch  the  disk,  they  will  be 
found  to  adhere.  This  adhesion  may  be  overcome  by  adding 
weights  to  the  opposite  arm  of  the  balance  ;  or  rather  the  disk 
may  be  drawn  away  from  the  mass  of  liquid,  for  it  will  still  carry 
with  it  a  film  of  the  liquid  ;  and  the  force  exerted  by  the  addi- 
tional weight  does  not  overcome  the  cohesive  force,  but  only  the 
attraction  of  aggregation  that  exists  between  the  particles  of  the 
liquid. 

391.  Phenomena  due  to  the  same  cause  are  observed  in  a  va- 
riety of  cases.     Thus :  the  surface  of  water  or  alcohol,  in  a  glass 
vessel,  is  slightly  raised  around  the  edges;  if  the  glass  be  di- 
minished in  size,  the  elevation  of  the  fluid  at  the  edge  will  in- 
crease ;  and  in  a  tube  of  glass  of  small  diameter,  a  column  will 
be  supported  within  it,  when  plunged  in  a  liquid,  above  the  level 
of  the  general  surface.     In  tubes  of  very  small  diameter,  called, 
from  their  size,  Capillary  Tubes,  this  column  may  amount  to  some 
inches. 

In  other  cases,  a  depression  exists  within  the  tube  ;  thus,  when 
one  of  glass  is  immersed  in  mercury,  that  liquid  will  be  obviously 
lower  within  the  tube  than  it  is  without. 

In  cases  where  the  liquid  is  raised  in  a  tube,  its  upper  surface 
assumes  a  concave  form,  which,  in  small  tubes,  differs  but  little 
from  a  portion  of  a  sphere  ;  and  in  cases  where  a  liquid  is  de- 
pressed, the  upper  surface  is  convex. 

Similar  phenomena  occur  between  two  tubes  of  different  di- 
ameters, placed  one  within  the  other,  and  between  two  parallel 
plates.  If  two  plates  are  inclined  to  each  other,  and  meet  at  one 


Book  V.]  COHESION.  391 

of  their  edges,  the  liquid  will  rise  between  them,  its  surface  as- 
suming the  form  of  a  curve. 

Two  solid  bodies  that  would  not  otherwise  adhere,  may  be 
made  to  stick  together  with  considerable  force,  by  this  action. 
Thus,  if  two  plates  of  glass  be  moistened  with  water,  aiyi  then 
pressed  together,  they  require  a  considerable  effort  to  separate 
them  ;  and  two  plates  of  polished  brass  may  be  in  the  Same  man- 
ner united  by  oil  or  melted  tallow.  In  the  latter  cas^,  the  in- 
terposed substance  becomes  solid  on  cooling,  and  the  force  with 
which  it  resists  an  effort  to  separate  the  plates,  becomes  very 
great.  In  general  terms,  when  two  solid  bodies  are  matte  to  co- 
here by  the  intervention  of  a  substance  that  can  be  applied  in  a 
"liquid  state,  and  which  afterwards  becomes  solid,  the  cohesion  is 
rendered  more  intense.  This  principle  is  applied  in  the  process 
of  soldering  the  metals. 

392.  In  order  to  examine  the  phenomena  of  the  rise  of  liquids 
in  capillary  tubes  : 

Let  us  suppose  a  prismatic  tube,  standing  in  a  vertical  posi- 
tion ;  that  its  sides  are  perpendicular  to  its  base  ;  and  that  it  is 
supported  in  a  vessel,  in  such  a  manner  that  it  plunges  at  its  base 
into  a  liquid  of  such  a  nature  as  to  rise  in  the  tube  above  its  na- 
tural level. 

The  attraction  of  the  tube  has  a  very  limited  sphere  of  action, 
for  the  height  to  which  fluids  rise  in  tubes  of  different  thickness, 
provided  their  interior  diameters  be  the  same,  is  constant.     Any 
small  column  of  the  liquid,  situated  in  or  near  the  axis  of  the  tube, 
will  not  be  affected  by  this  attraction,  but  must  be  supported  by 
the  action  of  the  adjacent  columns  of  fluid.    It  is  therefore  clear, 
that  the  action  of  the  tube  upon  the  column,  immediately  in  contact 
with  it,  is  the  final  cause  of  the  elevation  of  the  whole  mass.     A 
ring  of  the  fluid  is  first  raised  ;  this  raises  a  st^pnd  ;   the  second 
a  third,  and  so  on,  until  the  weight  of  the  fluid  exactly  balances 
the  attractive  forces  that  are  exerted  by  the  sides  of  the  tube.    In 
order  to  determine  the  conditions  of  equilibrium,  let  us  conceive 
the  tube  to  be  produced  in  the  form  of  a  syphon,  by  a  part  of  no 
thickness,  and  which,  therefore,  does  not  by  its  attraction  inter- 
fere with  the  conditions  of  equilibrium,  and  does  not  prevent  the 
re-action  of  the  fluid  partieles  contained  within  it  upon  those  in 
the  tube.     It  is  obvious  that  the  column  contained  within  this 
imaginary  tube,  will  exactly  replace,  in  its  fluid  action,  the  whole 
mass  contained  in  the  vessel ;  and  in  the  case  of  equilibrium,  the 
pressure  in  the  two  branches  of  the  tube  must  be  identical.     But 
the  columns  of  fluid  in  the  two  branches  are  of  unequal  heights  ; 
the  difference  of  pressure  that  results  from  this  inequality,  must, 
therefore,  be  counteracted  by  the  attractions  of  the  prism  and  the 
fluid  ;  these  are  exerted  in  a  vertical  direction  in  the  original 
branch  of  the  tube. 


392  ATTRACTION  OF  [J3ook   V. 

As  the  prism  is  assumed  to  be  vertical,  its  base  is  horizontal. 
The  fluid  contained  in  the  additional  tube,  is  attracted  vertically 
downwards:  (l,)  by  its  own  particles ;  (2,)  by  the  fluid  that  sur- 
rounds it.  But  these  two  attractions  are  counteracted  by  the  like 
attractions  that  the  fluid  contained  in  the  second  vertical  branch 
sustains.  The  fluid  of  the  vertical  branch  of  the  second  tube  is 
besides  attracted  vertically  upwards  by  the  fluid  in  the  first  tube. 
Bui  this  attraction  is  destroyed  by  the  attraction  the  former  exerts 
upon  the  latter  column  of  fluid.  These  reciprocal  attractions 
may  therefore  be  disregarded. 

Finally  the  fluid  in  the  second  tube  is  attracted  vertically  upwards 
by  the  first  tube  ;  hence  there  results  a  vertical  force  that  contri- 
butes to  destroy  the  excess  of  pressure  due  to  the  elevation  of  the 
fluid  in  the  first  tube.  This  force  tends  to  destroy  the  excess  of 
pressure  exerted  in  opposition  to  it,  by  the  column  of  fluid  raised 
in  the  original  tube  above  the  general  level. 

This  force  we  shall  call  P. 

The  forces  that  act  upon  the  fluid  contained  in  the  original 
tube,  are  as  follows  : 

(1.)  It  is  attracted  by  itself;  but  as  the  reciprocal  attractions  of 
the  particles  of  a  solid  body  do  not  cause  any  motion,  we  may  ab- 
strKct  this  attraction,  for  we  may,  in  a  tube  of  uniform  bore,  stand- 
ing in  a  vertical  position,  consider  the  vertical  pressure  as  if  it 
were  produced  by  a  solid  body  filling  the  tube. 

(2.)  The  elevated  fluid  is  attracted  downwards  by  the  liquid  co- 
lumn contained  in  the  part  beneath  the  level  of  the  external  fluid  ; 
but  it  attracts  in  its  turn  with  an  equal  force,  and  these  attractions 
mutually  destroy  each  other. 

(3.)  The  fluid  is  also  attracted  downwards  by  the  fluid  that  sur- 
rounds the  ideal  prolongation  of  the  tube ;  hence  there  results 
a  vertical  force  directed  downwards,  that  we  shall  call  —  P',  its 
sign  being  negate,  in  order  to  represent  that  its  direction  is  op- 
posite to  thf»<  of  the  force  P.  The  forces  would  be  exactly  equal 
if  the  tpi>e  were  composed  of  the  same  material  with  the  fluid. 
Thuir  inequality  is  therefore  due  to  the  difference  in  the  intensity 
of  the  attractive  forces  exerted  by  the  particles  of  the  fluid  upon 
each  other,  and  by  the  particles  of  the  tube  upon  those  of  the  fluid. 
If  we  take  r  and  r'  to  represent  these  respective  intensities,  we 
have 

P  :  P'  :  :  r  :  r'.  (398) 

(4. )  The  fluid  in  the  first  tube  is  attracted  vertically  upwards  by 
the  matter  of  the  tube,  and  this  force  is  obviously  equal  to  P. 

The  whole  of  the  attractive  forces  that  act,  are,  therefore, 

2  P— P',  (399) 

and  they  arc  equal  to  the  weight  of  the  column  that  they  raise. 

If  a-  represent  the  force  of  gravity,  D  the  density  of  the  fluid, 
and  V  the  volume  of  the  elevated  column, 

g  DV=2P— P'.  (400) 


Book  J7".]  COHESION.  393 

It  is  obvious  from  this  equation,  that  the  quantity  V,  will  always 
have  the  same  sign  with  the  quantity  2  P  —  P'. 
Hence  : 

393.  When  the  attraction  of  the  particles  of  the  fluid  for  each 
other  is  half  that  which  the  particles  of  the  tube  have  for  those 
of  the  fluid,  the  level  within  and  without  the  tube  will  be 
the  same  ;  when  the  former  is  less  than  half  the  latter,  the  fluid 
will  be  raised  ;  but  when  it  is  greater,  the  surface  of  the  fluid  will 
be  depressed. 

The  action  being  exerted,  at  imperceptible  distances,  will  only 
be  felt  by  the  columns  in  immediate  contact  with  the  tube  ;  we 
may,  in  consequence,  neglect  the  curvature  of  the  tube,  and  con- 
sider it  as  developed  into  a  plane  surface. 

The  attractive  force,  P,  will  therefore  be  proportioned  to  the 
size  of  this  plane,  and  may  be  represented  by  r  C,  C  being  the 
contour  of  the  surface.  And  for  a  like  reason, 

P'=r'C; 
whence 

g-DV  =  (2r—  r')C.  (401) 

This  formula  is  applicable  to  all  the  cases  that  have  been  ob- 
served in  practice,  and  the  results  that  flow  from  it  are  consistent 
with  observation. 

In  cylindrical  tubes  : 

Let  a  be  the  radius  of  the  interior  of  the  tube,  h  the  height  of 
the  column,  measured  from  the  level  of  the  fluid  without,  to  the 
curved  surface  it  supports  or  depresses.  The  volume  of  this 
column  is 

*a?h.  (402) 

To  this,  must  be  added  the  volume  of  the  meniscus  in  which 
it  terminates,  which  will  be  the  difference  between  the  volume  of 
a  cylinder,  whose  height  and  base  are  both  equal  to  a,  and  the 
hemisphere  whose  radius  is  a,  or  to 

2*a?  tea3 

to?  —  —  g  —  ,  or  simply  to  -g~. 

The  whole  volume  of  fluid  raised,  will  therefore  be 


and  the  contour  of  the  base  C,  is 


substituting  these  values  in  the  foregoing  equation,  we  have 

g-D  («a2h+*a3)  =  (2r—  r')  2«a  ;  (403) 

and  dividing  by  ta, 

(-*     a\         /2r  —  r'\ 
ft  +  3)=2(-Tcr)  <404> 

50 


394  ATTBAOTION  OF  [Book   V- 

If  the  tubes  be  of  the  same  material,  and  be  plunged  in  the 
same  fluid,  whose  temperature  is  constant,  r,  ^^  sf  and  D  will  be 
the  same  in  every  case,  and  the  second  member  of  the  equation 
will  be  a  constant  quantity,  which  we  may  call  A,  and 


a     /*+       =A.  (405) 

whence 

a     A 
M-3--.  (406) 

When  the  tube  is  small,  the  height,  ft,  is  great,  compared  with 

a 
the  radius  a,  and  the  quantity  ~,  may  be  neglected,  or  will  be 

masked  by  the  errors  of  observation  ;  we  may  therefore  assume, 

h=~  (407) 

Hence  : 

394.  In  capillary  tubes,  liquids  will  rise  or  be  depressed,  ac- 
cording to  the  relation  between  the  attractive  forces  stated  in  the 
preceding  section,  to  heights  inversely  proportioned  to  the  di- 
ameters of  the  tubes. 

Applying  the  formula  (402),  by  a  similar  method,  to  the  case 
of  two  parallel  solid  surfaces,  at  a  small  distance  from  each  other, 
which  we  shall  call  J,  we  would  obtain 

A=-j.  (408) 

Hence  : 

A  liquid  will  be  raised  or  depressed  between  two  parallel 
plates,  at  small  distances  from  each  other,  to  heights  that  are 
inversely  proportioned  to  the  distances  between  the  plates  ; 
and  these  heights  will  be  the  same  as  those  to  which  the  same 
liquid  would  rise  in  a  tube  of  the  same  material  with  the  plates, 
whose  radius  is  equal  to  the  distance  between  the  plates. 

Two  concentric  tubes,  or  a  tube  surrounding  a  solid  cylinder, 
may  be  considered  as  a  case  of  this  kind  ;  the  liquid  will  rise  be- 
tween them  to  half  the  height  it  would  in  a  tube  whose  diameter 
is  equal  to  their  distance. 

Such  is  the  theory  of  Laplace,  which,  neglecting  certain  small 
quantities,  coincides  with  the  observations  of  former  experi- 
menters, but  which  by  the  more  accurate  experiments  of  Gay  Lus- 
sac,  has  been  found  to  be  true,  even  in  its  full  extent  ;  the  refine- 
ments introduced  by  tbe  latter,  having  enabled  him  to  detect  the 
small  variations  in  level  that  had  escaped  those  who  preceded 
him. 

Laplace  has  also  investigated  the  nature  of  the  surface  that 


Book  V.*\  COHESION.  395 

would  be  assumed  by  a  liquid.  We  have  not  spa^ce,  nor  would 
it  be  consistent  with  an  elementary  treatise,  to  enter  into  his 
beautiful  and  complete  analysis;  we  shall,  therefore,  content 
ourselves  with  stating  the  results  of  observation  which  are  con- 
firmed by  the  theory. 

In  tubes  of  small  diameter,  the  surface  of  the  liquid,  whether 
elevated  or  depressed,  is  always  spherical ;  but  in  the  former  case, 
it  is  a  concave  portion,  in  the  latter,  a  convex  one,  of  a  sphere. 

395.  Aeriform  fluids  are  also  affected  by  capillary  attraction  ; 
and  this  is,  in  some  cases  so  intense,  as  to  condense  them  into  a 
volume  considerably  less  than  that  which  they  occupy  under  or- 
dinary pressures.  The  most  remarkable  instance  of  this  sort  is, 
the  absorption  of  the  gases  by  charcoal.  This  substance,  when 
recently  burnt,  takes  up  within  twenty-four  hours,  according  to 
the  experiments  of  Saussure,  of 

Ammonia         .  .         90  times  its  volume. 
Muriatic  Acid  Gas  85       " 

Sulphurous  Acid  65       " 

Sulphuretted  Hydrogen     55       " 
Nitrous  Oxide  40       " 

Carbonic  Acid  .         35       " 

OlefiantGas  .         35       " 
Oxygen            .  9.25  « 

Nitrogen          .  .  7.5    " 

Hydrogen        .  .  1.75 " 

Solutions  also,  generally  ranked  among  chemical  phenomena, 
are  unquestionably  due  to  the  mechanical  action  of  the  particles 
of  the  solvent  upon  those  of  the  solid  ;  and  this  attraction  is  op- 
posed by  the  attraction  of  aggregation  that  the  particles  of  the 
latter  have  for  each  other.  The  mixture  of  liquids,  particularly 
when  attended  with  the  diminution  of  bulk  called  Concentration, 
may  be  also. included  among  the  cases  of  cohesive  attraction. 


;  rAi, 


BOOK   VI. 


OF  THE   MOTION   OF  FLUIDS. 


•AS'p&Wno  G;J/  '  • 

CHAPTER  I. 

THEORY  OF  THE  MOTION  OF  LIQUIDS. 

396.  We  may,  by  the  application  of  the  principle  of  D'Alem- 
bert  to  the  equations  of  the  equilibrium  of  fluids,  deduce  the  gene- 
ral equations  of  their  motion.  These  equations  are,  however,  ex- 
tremely complicated,  and  are  incapable  of  complete  integration. 
It  has,  therefore,  been  more  usual  to  proceed  by  means  of  an  hy- 
pothesis, that  appears  at  first  sight  to  approximate  to  the  truth. 
This  hypothesis  considers  the  fluid  to  be  divided  into  a  number 
of  horizontal  layers  or  strata,  and  that  the  particles  in  each  of 
these  separate  layers  have  a  common  velocity.  Each  of  the 
layers  then  would  continue  parallel  to  itself,  and  is  composed  of 
the  self-same  particles  throughout  the  whole  duration  of  its  mo- 
tion ;  and  any  given  layer  will  descend  and  occupy  the  place  of 
that  which  is  immediately  beneath  it,  and  so  in  succession. 

If  this  hypothesis  be  applied  to  the  case  of  a  liquid  contained 
in  a  vessel  of  irregular  figure,  it  might  be  demonstrated,  that  in 
moving  through  it,  each  different  layer  will  have  a  velocity  in- 
versely proportioned  to  the  area  of  the  section  of  the  vessel  at 
which  this  layer  is  situated. 

This  hypothesis  is,  however,  in  many  cases,  at  utter  va- 
riance with  what  is  observed  in  practice.  When,  for  instance, 
a  liquid  is  placed  in  a  prismatic  vessel  that  is  permitted  to  dis- 
charge itself  through  an  orifice  in  the  bottom,  the  surface  ceases 
at  once  to  be  level,  being  depressed  immediately  above  the  ori- 
fice. The  surface,  therefore,  becomes  concave,  and  the  particles 
that  compose  it,  instead  of  tending  to  descend  in  a  horizontal  layer, 
appear  to  move,  as  it  were,  upon  an  inclined  plane,  to  the  point 
that  is  most  depressed,  and  thence  to  descend  vertically  to  the 
orifice.  Not  only  does  this  tendency  to  the  vertical  line  appear 


398  THEORY  OF  THE  [Book   V{. 

at  the  surface,  but  it  is  also  manifested  in  the  inferior  strata  :  thus. 
if  a  liquid  of  less  density  be  poured  upon  the  surface  of  the  liquid 
that  is  first  introduced  into  the  vessel,  it  speedily  joins  the  de- 
scending current  and  passes  out;  but  it  does  not  pass  out  unmix- 
ed, for  the  two  liquids  are  intimately  mingled,  until  the  whole  of 
the  lighter  be  discharged.  This  tendency  in  the  lower  strata, 
towards  the  column  that  is  immediately  above  the  orifice,  is  also 
manifested  by  placing  in  the  vessel  powders  of  a  density  equal  to 
that  of  the  liquid  ;  these  will  be  seen  to  move  towards  the  orifice, 
in  curves  of  various  degrees  of  convergence,  and  will  unite  them- 
selves with  the  effluent  stream. 

397.  These  phenomena  may  be  accounted  for,  by  supposing 
that  each  particle  of  the  liquid  descends  towards  Ihe  orifice  ex- 
actly as  if  it  were  unconnected  with  the  surrounding  mass.  It 
would  therefore  acquire  an  uniformly  accelerated  velocity  ;  the  co- 
lumn would  be  broken,  and  spaces  left  between  the  particles,  to- 
wards which  the  pressure  of  the  adjacent  liquid  would  impel  other 
particles,  that  would  thus  join  in  the  current,  and  occupy  the 
void  spaces  of  the  column. 

If  the  orifice  be  pierced  in  the  side  of  the  vessel,  the  particles 
of  liquid  will  still  move  towards  it  like  falling  bodies,  but  will 
describe  a  curve  instead  of  a  vertical  line.  If  the  orifice  be 
made  in  a  part  of  the  vessel  that  will  permit  it  to  be  directed  up- 
wards, the  particles  will  again  reach  it,  under  circumstances 
similar  to  those  which  are  found  in  bodies  moving,  under  the  ac- 
tion of  gravity,  upon  curved  surfaces  whose  tangents  make  with 
a  horizontal  line,  the  same  angle  that  the  direction  of  the  orifice 
makes  with  the  horizontal  plane. 

When  the  motion  begins,  the  particles  immediately  in  contact 
with  the  orrfice,  move  from  a  state  of  rest ;  and  those  that  lie  be- 
tween them  and  the  surface  cannot  be  accelerated  without  accele- 
rating those  beneath  them  ;  thus  a  resistance  will  be  opposed  to 
the  descent  that  will  for  a  time  prevent  the  particles  that  proceed 
from  the  surface  of  the  liquid,  from  acquiring  the  velocity  a  falling 
solid  body  would  attain  in  passing  through  the  same  space.  The 
time  for  which  this  resistance  will  produce  an  appreciable  effect  is 
but  short,  for  so  soon  as  the  first  particle  that  issues  shall  have 
fallen  through  a  space  equal  to  the  altitude  of  the  fluid  above  the 
orifice,  its  velocity  will  become  equal  to  that  of  the  particles  pro- 
ceeding from  the  surface,  would  acquire  at  the  orifice ;  and  it  will 
no  longer  retard  the  column  that  follows  it. 

If  the  fluid  move  in  a  vessel  of  variable  section,  its  velocity  does 
not  vary  with  the  area,  but  there  is  a  column  or  vein  that  moves 
in  it,  precisely  as  if  it  were  in  a,  prismatic  vessel,  while  in  the 
parts  whose  areas  are  greatest,  eddies  are  formed.  The  nature 


- 

Book  J7/.]  MOTION  OF  LIQUIDS.  399 

and  character  of  these  eddies  will  be  more  particularly  considered 
hereafter. 

The  vein  that  moves  according  to  the  law  of  gravity,  will  be 
resisted  by  the  viscidity  of  the  neighbouring  particles  of  the  li- 
quid ;  it  will  also  be  retarded,  in  consequence  of  the  particles 
that  join  it  in  its  course  having  a  less  velocity  than  those  which 
proceed  from  a  higher  level.  The  former  will  receive  a  part  of 
the  motion  of  the  latter,  and  the  whole  will  move  forward  with  a 
common  velocity.  If  the  vein  nearly  fill  up  the  vessel  in  which 
it  is  moving,  it  will  meet  with  a  resistance  analagous  to  friction 
from  the  sides,  and  will  be  influenced  by  the  attraction  of  cohe- 
sion. 

If  these  circumstances  be  left  out  of  account,  we  may  consider 
the  velocity  of  any  part  of  a  vein  of  a  gravitating  liquid  in  mo- 
tion, to  be  such  as  would  be  due  to  the  height  of  the  surface  of 
the  liquid,  above  the  point  whose  motion  is  considered.  In  some 
cases,  the  circumstances  to  which  we  have  referred  are  of  no  mo- 
ment, and  may  safely  be  neglected  ;  in  others,  they  affect  the  ve- 
locity in  a  high  degree,  and  even  render  constant  that  motion 
which  would  otherwise  be  uniformly  accelerated. 

It  therefore  becomes  necessary  to  distinguish  different  cases 
of  the  motion  of  liquids.  The  more  important  of  these  cases  are 
five  in  number  :  viz., 

(1.)  The  motion  of  liquids  that  issue  from  orifices  pierced  in 
thin  plates. 

(2.)  The  motion  of  liquids  through  orifices  cut  in  thick  plates, 
or  through  short  tubes  adapted  to  orifices. 

(3.)  The  motion  of  liquids  in  long  tubes  or  pipes. 

(4.)  The  motion  of  liquids  in  open  channels. 

(5.)  The  motion  of  liquids  over  the  edges  of  the  lower  portion 
of  the  irregular  sides  of  the  vessel  or  basin  that  contains  them. 

We  shall  take  up  the  consideration  of  these  several  cases  in  the 
order  in  which  they  have  been  mentioned. 


400  .  MOTION  OF  [Book  VI. 


CHAPTER  II. 

OF  THE  MOTION  OF  LIQUIDS  THROUGH  ORIFICES  PIERCED  IN  THIN 

PLATES. 

398.  If  we  abstract  the  circumstances  of  which  we  have  spo- 
ken in  the  close  of  the  last  chapter,  a  liquid,  on  reaching  an  orifice, 
will  have  a  velocity  due  to  the  height  of  the  level  of  the  liquid 
above  the  orifice.  To  represent  this  in  a  formula  : 

Let  »  be  the  velocity  with  which  the  liquid  issues  ;  h  the  verti- 
cal height  of  the  surface  of  the  liquid  above  the  orifice  ;  g  the 
measure  of  the  force  of  gravity  ;  then  by  (61), 

(409) 


From  this  formula  a  variety  of  consequences  immediately  fol- 
low. 

(1.)  If  the  vessel  be  kept  constantly  full,  the  velocities  of  the 
effluent  fluids,  from  any  orifice  given  in  position,  is  constant. 

(2.)  From  orifices  pierced  at  different  heights  in  the  side  of  a 
vessel  kept  constantly  full,  the  velocities  are  as  the  square  roots 
of  the  depths  of  the  orifices  beneath  the  level  surface  of  the 
liquid. 

(3.)  If  the  vessel  be  permitted  to  empty  itself,  the  velocity 
with  which  the  liquid  will  issue  from  a  given  aperture  is  equally 
retarded. 

(4.)  As  none  of  the  circumstances  of  which  we  have  spoken 
will  affect  the  area  of  the  column  of  liquid,  discharged  from  a 
given  orifice,  the  quantities  discharged  in  the  elements  of  the 
time  will  be  directly  proportioned  to  the  velocities. 

Therefore,  as  in  a  vessel  permitted  to  empty  itself,  the  velocities 
are  uniformly  retarded,  it  is  obvious  from  §  49,  that  twice  as 
much  liquid  should  flow  from  a  given  orifice,  in  the  unit  of  time, 
when  the  vessel  is  kept  constantly  full,  as  should  flow  from  the 
same  orifice  in  the  same  time,  when  the  vessel  is  permitted  to 
empty  itself;  and  so,  when  the  vessel  is  permitted  to  empty  it- 
self, it  should  occupy  twice  as  much  time  to  discharge  a  given 
quantity,  as  would  suffice  for  an  equal  discharge  from  a  vessel 
kept  constantly  full. 

(5.)  In  emptying  a  vessel  through  an  orifice  in  its  bottom,  the 
quantities  discharged  in  equal  times  should  decrease  as  the  series 
of  odd  numbers. 


Book  VI ^ 


LIQUIDS. 


401 


(6.)  When  a  fluid  issues  from  a  vessel  in  a  vertical  direction, 
it  will  rise  or  fall  in  a  vertical  line  ;  and  if  it  be  directed  upwards, 
as  it  has  an  initial  velocity  due  to  the  height  of  the  surface  of  the 
liquid  above  the  orifice,  it  should  rise,  if  we  abstract  from  the  re- 
sistances it  meets,  to  the  level  of  that  surface.  The  velocity  in 
the  jet  will  be  uniformly  retarded. 

(7.)  If  a  liquid  spout  from  an  orifice  in  any  other  than  a  verti- 
cal direction,  the  joint  action  of  its  effluent  motion,  with  a  velo- 
city due  to  the  depth,  and  of  the  force  of  gravity,  would,  if  no 
other  forces  acted,  cause  it  to  describe  a  parabola,  whose  directrix 
will  lie  in  the  horizontal  plane,  coinciding  with  the  surface  of 
the  liquid,  (  §53). 

(8.)  If  different  orifices  be  pierced  in  a  horizontal  direction,  in 
the  vertical  sides  of  a  vessel,  which  is  kept  constantly  filled  with 
a  liquid,  the  effluent  velocities  of  each  vein  of  liquid  will  vary, 
and  each  will  have  a  different  distance  to  descend  before  it  reaches 
the  horizontal  plane  that  coincides  with  the  bottom  of  the  ves- 
sel. Hence  the  curves  described  in  each  case  will  be  different. 

The  comparative  distances  to  which  the  several  jets  will  pass 
over  this  horizontal  plane  may  be  thus  investigated  : 

Let  AB  be  the  side  of  a  prismatic  vessel,  at  the  point  D,  in 
which  an  orifice  is  pierced,  whence  a  fluid  kept  constantly  at 
the  level  of  the  point  A  issues  in  a  horizontal  direction.  Bisect 
AB  in  C,  and  around  C  describe  a  semicircle ;  draw  the  ordinate 
DE  through  the  point  D. 


51 


! 


402  MOTION  OF  [Book  VI. 

Let 


DB=a, 
DE=*. 

The  liquid,  in  issuing  from  the  point  d,  will  have  a  velocity  due 
to  the  height  /i,  and  equal  to 


this  velocity  would  carry  it  with  uniform  velocity  through  twice 
the  distance  /i,  in  the  same  space  of  time  that  it  has  taken  to  ac- 
quire that  velocity.  But  so  soon  as  it  leaves  the  orifice,  it  has  its 
direction  changed  and  describes  a  parabola.  Under  the  action 
of  this  deflecting  force,  it  will  reach  the  horizontal  plane  BF,  in 
the  same  time  that  it  would  have  fallen  through  the  height  DB, 
if  it  were  influenced  by  the  force  of  gravity  alone.  The  times  of 
describing  AD,  and  DB,  are  respectively 

2h  2a 

V~,  and  V-> 

in  the  first  of  these  times,  the  projectile  force  would  carry  it  with 
uniform  motion  through  the  space  2ft,  and  in  the  second,  through 
•  the  distance  at  which  the  jet  of  fluid  strikes  the  horizontal  plane, 
which  we  shall  call  d.  As  the  spaces  are  as  the  times,  we  have 
the  analogy 

2h         2a 
V—  :  V—  ::2h:d, 

o  6 

whence  we  obtain  for  the  value  o 


The  distance  then  from  the  perpendicular  side  of  the  vessel  at 
which  the  parabolic  jet  strikes  the  horizontal  plane  on  which  the 
vessel  stands,  is  equal  to  twice  the  line  ordinately  applied  to  a  se- 
micircle, of  which  the  vertical  depth  of  the  fluid  is  the  diameter, 
From  the  centre  of  the  circle,  or  half  the  height  of  the  liquid, 
the  horizontal  range  will  be  the  greatest  ;  and  at  equal  distances 
above  or  below  this  point,  the  horizontal  ranges  are  equal. 

399.  When  a  liquid  issues  from  an  orifice  with  a  given  velo- 
city, whether  it  be  such  as  is  determined  from  the  foregoing  ab- 
stract theory,  or  modified  by  physical  circumstances,  it  might  be 
at  first  sight  concluded  that  the  quantity  discharged  in  the  linit 
of  time,  might  be  determined  by  multiplying  the  area  of  the  ori- 
fice by  the  velocity.  This,  however,  is  not  the  case.  The  liquid 
does  not  issue  in  a  prismatic  shape,  and  hence  its  quantity  is  not 
measured  by  the  contents  of  a  prism  of  which  the  orifice  is  the 
base.  On  the  contrary,  the  vein  of  liquid  is  obviously  contracted 


Book  VL]  '   LIQUIBS.  403 

soon  after  it  issues  forth,  and  again  spreads  out  to  dimensions 
larger  than  those  of  the  orifice.  That  this  ought  to  be  the  case, 
will  be  understood  from  reference  to  the  foregoing  theory.  For, 
as  the  particles  move  from  the  lower  layers  of  the  liquid  to  join 
the  vein  directed  towards  the  orifice,  they  have  a  motion  oblique 
to  that  of  the  general  current  ;  but  from  the  particles  that  compose 
it,  they  receive  a  change  of  direction,  and  at  the  same  time  re-act 
upon  them.  Thus  the  particles  that  issue  from  the  edges  of  the 
orifice,  will  have  a  direction  inclined  to  the  axis  of  the  jet,  and 
the  stream,  of  which  they  form  a  part,  must  contract  in  dimen- 
sions. As  these  directions  will  cross  each  other,  some  of  the  par- 
ticles will,  below  the  point  to  which  they  converge,  be  forced 
outward  from  the  axis.  Thus  the  shape  of  the  jet  will  be  one 
formed  of  two  truncated  conoidal  frusta;  one  of  these  will  have 
the  orifice  for  its  base,  and  a  definite  altitude  ;  while  the  other 
will  have  the  smaller  base  of  the  former  for  its  lesser  base,  and 
an  unlimited  altitude.  The  investigation  by  analytic  means  of 
the  exact  figure  of  these  conoidal  frusta,  were  it  practicable, 
would  yet  be  attended  with  no  valuable  results  ;  for  the  contrac- 
tion in  the  vein  is  connected  with  the  change  in  velocity  grow- 
ing out  of  the  viscidity  of  the  liquid,  and  the  mutual  action  of  its 
particles  ;  therefore,  the  separate  effects  of  these  tw>o  different 
actions  cannot  be  distinguished  in  experiment.  We,  in  conse- 
quence, consider  that  the  true  velocity  is  given  by  the  principles 
of  the  preceding  section,  and  that  the  whole  diminution  in  the 
quantity  discharged,  is  due  to  the  contraction  of  the  vein. 

The  contraction  that  occurs  in  the  form  of  a  jet  of  fluid,  issuing 
from  an  orifice,  is  too  apparent  to  have  escaped  notice,  even  at 
an  early  period.  Newton,  however,  was  the  first  who  attempted 
to  ascertain  the  amount  of  this  contraction.  In  this  he  was  not 
perfectly  successful,  but  conceived  that  he  had  found  it  to  be  in 
the  ratio  5  :  7,  or  of  \/l  :  \/2. 

If  a  be  the  area  of  the  orifice,  q  the  quantity  discharged,  and 
we  use  the  same  notation  as  before, 


and  upon  the  hypothesis  that  the  quantity  is  found  by  multiplying 
the  area  of  the  orifice  by  the  velocity, 

q=aV2gh.  (410) 

This  is  called  the  Theoretic  Discharge.  If  it  be  reduced  in  the 
ratio  given  by  Newton,  we  have 

q=aVgh.  (411) 

As  the  change  in  this  formula  is  made  in  the  second  part  of  it 
which  represents  the  velocity,  a  false  inference  has  been  drawn 


404  MOTION  OP  [Book  VI. 

by  some  writers,  who,  forgetting  the  circumstance  of  the  con- 
traction of  the  vein,  have  stated  that  the  velocity  itself  is  dimin- 
ished, and  becomes  no  more  than  is  due  to  half  the  height  of  the 
fluid  above  the  orifice.  'This,  however,  is  an  obvious  error;  the 
velocity  is  but  little  affected  when  the  liquid  issues  from  an  ori- 
fice pierced  in  a  thin  plate,  and  the  diminution  in  the  actual  dis- 
charge compared  with  the  theoretic,  is  principally  due  to  the 
contraction  of  the  vein. 

The  best  experiments  on  the  phenomena  of  liquids  issuing 
from  orifices  pierced  in  thin  plates,  are  those  of  Bossut.  From 
these  it  appears, 

( 1 ).  That,  except  when  a  vessel  is  nearly  exhausted,  no  sensible 
error  can  arise  from  considering  the  velocities  as  due  to  the  height 
of  the  level  surface  of  the  liquid  above  the  orifice. 

The  narrowest  part  of  a  jet  of  liquid,  issuing  from  an  orifice, 
is  called  the  Vena  C on  tract  a  ;  this  is  situated  at  a  distance  from 
the  orifice,  when  of  a  circular  figure,  that  is  equal  to  its  radius. 

(2).  The  figure  of  a  vertical  jet,  lying  between  a  circular  ori- 
fice and  the  vena  contracta,  is  nearly  a  conic  frustum,  whose  two 
bases  have  to  each  other  the  ratios  of  62  :  100,  or  nearly  as  5  :  8, 
instead  of  5  :  7,  as  stated  by  Newton. 

The  figure  of  the  jet,  beyond  the  vena  contracta,  is  also  sensi- 
bly a  conic  frustum,  the  angle  of  whose  vertex  is  32°. 

In  orifices  of  any  other  figure,  the  same  ratio  is  nearly  true  be- 
tween the  dimensions  of  the  orifice  and  the  section  of  the  liquid 
at  the  vena  contracta  ;  but  the  figure  of  the  jet  is  not  pyramid- 
ical ;  at  a  small  distance  from  the  orifice,  if  of  a  rectangular  shape, 
it  assumes  the  form  of  a  cross,  whose  arms  He  in  the  direction  of 
the  diagonals  of  the  orifice;  beyond  this,  the  section  again  be- 
comes rectangular,  with  diagonals  parallel  to  the  sides  of  the  ori- 
fice, and  this  figure  is  retained  in  the  subsequent  enlargement  of 
the  vein. 

Analogous  changes  of  figure  take  place  in  the  section  of  the 
vein  of  liquid,  when  the  orifice  has  the  figure  of  a  triangle  or  a 
polygon. 

(3).  So  long  as  the  vessel  continues  to  hold  a  column  of  liquid  at 
a  considerable  height  above  the  orifice,  the  actual  discharges  from 
orifices  of  any  figure  whatsoever,  are  to  the  theoretic  nearly  in 
the  ratio  ffa ;  and  the  velocities  and  quantities  are  nearly  pro- 
portioned to  the  square  roots  of  the  depth  of  the  liquid. 

(4).  Small  orifices  discharge  rather  less  than  the  reduced  quan- 
tity, large  orifices  rather  more  ;  and  of  orifices  of  equal  area  and 
unequal  circumferences,  those  with  the  smallest  circumference 
discharge  the  greatest  quantity. 


Book  VL]  LIQUIDS.  405 

400.  When  the  liquid  spouts  vertically  upwards,  there  is  a  de- 
viation from  the  theoretic  height  that  can  be  easily  perceived. 
That  this  should  be  the  case,  will  be  obvious  when  we  consider 
that  the  particles  of  liquids  are  retarded  by  the  column  that  has 
preceded  them,  the  particles  of  which  are  moving  with  a  con- 
tinually diminishing  velocity  ;  and  that  the  particles,  after  reach- 
ing; their  utmost  height,  tend  to  return  in  a  vertical  direction:  a 
part  of  the  force  of  the  ascending  column  must  therefore  be  ap- 
plied to  force  them  to  one  side.  In  consequence  of  the  latter 
circumstance,  it  has  been  found  that  a  jet  of  liquid,  when  slightly 
inclined,  rises  higher  than  if  pointed  vertically  upwards.  When 
the  original  velocity  is  due  to  a  great  height,  and  is,  in  conse- 
quence, large,  the  resistance  of  the  air  becomes  a  powerful  re- 
tarding force,  and  hence  creates  a  limit  beyond  which  no  head  of 
water,  however  great,  can  cause  a  vertical  jet  of  liquid  to  rise. 
The  height  to  which  a  liquid  rises  vertically  upwards,  and  which 
is,  therefore,  always  lower  than  the  level  of  the  liquid  in  the 
vessel  whence  it  issues,  is*  called  the  Effective  Head. 

It  has  been  ascertained  by  the  experiments  of  Mariotte,  that 
a  head  of  five  French  feet  and  an  inch,  produces  a  vertical  jet  of 
five  feet. 

If  H,  and  H',  be  the  actual  heads  of  two  masses  of  water  that 
form  vertical  jets  ;  fe,  and  A',  the  effective  hea'ds,  the  experiments 
of  Mariotte,  give  the  following  relation  between  them. 
H—  h         h? 


and 

H-(H'—  /O^+fc,  (413) 
or  taking  the  above  data,  where 

,      .  '  H'=5T2-     •  ,  •  -•"  ••!  ;aS  •>, 

h'=5 

A;  '             •      -(414) 


whence  the  real  head  that  will  produce  a  jet  of  any  given  height, 
can  be  estimated. 

401.  If  the  velocity  due  to  the  effective  head  be  employed  in 
the  parabolic  theory,  instead  of  the  actual  velocity,  the  results 
will  be  nearly  identical  with  those  that  actually  take  place. 
Hence,  from  the  theory  of  projectiles,  §  250,  we  have  for  the 
height  to  which  an  inclined  jet  will  rise, 

hsin.  2i;  (415) 


406  MOTION  or  [Book  VI. 

and  for  the  horizontal  distance  to  which  it  will  reach, 

ZJisin.  2i.  (416) 

402.  The  rules  of  §  399,  are  only  found  to  hold  good  in  prac- 
tice when  the  height  of  the  column  of  liquid  in  the  vessel  is  large, 
compared  with  the  area  of  the  orifice ;  as  the  height  lessens  and 
the  vessel  becomes  nearly  empty,  the  velocity  of  discharge  is  di- 
minished below  that  due  to  the  height ;  and  the  contraction  of 
the  vein  increases.  This  grows  partly  out  of  a  rotary  motion 
that  often  takes  place  in  the  fluid  in  the  vessel,  and  causes  a  cen- 
trifugal force  that  lessens  the  action  of  gravitation.  The  parti- 
cles, therefore,  that  reach  the  orifice  in  a  vertical  direction,  have 
a  less  velocity  than  they  would  otherwise  acquire;  and  those 
whose  direction  is  oblique,  are  less  powerfully  acted  upon,  and 
continue  their  oblique  course  longer. 

The  formation  of  the  vortex,  whose  rotary  motion  produces 
these  effects,  may  be  thus  explained  : 

The  particles  that  enter  the  vein  directed  towards  the  orifice, 
have  a  motion  that  may  be  resolved  into  two,  one  in  the  vertical, 
the  other  in  a  horizontal  direction.  The  latter  is  obviously  a 
centripetal  force.  If  a  third  force  act  upon  any  one  of  the  par- 
ticles, in  any  other  direction  than  that  of  these  two  components, 
it  will  cause  the  particle  to  deviate  in  a  horizontal  direction  ; 
for  one  of  its  component*  will  be  horizontal,  and  thus  the  motion 
in  the  direction  of  the  radius  will  become  circular,  if  the  disturb- 
ing force  be  of  sufficient  intensity,  or  spiral,  if  it  be  less  intense. 
In  the  latter  case,  it  may  be  considered  as  taking  place  in  circles, 
successively  decreasing  in  magnitude,  and  the  laws  of  circular 
motion  of  §  64,  will  be  applicable.  This  disturbing  force  may 
proceed  from  extrinsic  causes,  but  it  may  arise  also  from  irregu- 
larity in  the  figure  of  the  vessel,  or  from  the  position  of  the  ori- 
fice being  in  any  other  point  than  the  centre  of  magnitude  of  a 
base  of  regular  figure;  for  in  either  of  these  cases,  the  particles 
that  join  the  vein  at  a  given  level,  will  reach  it  with  different 
inclinations  and  velocities,  and  will  therefore  effect  each  other's 
motions. 

If  we  call  the  velocity  of  rotation  of  any  particle  is  its  distance 
from  the  axis  of  the  vein,  r,  and  the  time  of  a  revolution,  /,  the 
velocity  of  rotation  will  be  constant ;  and  will,  therefore,  in  the 
successively  decreasing  circles,  be  inversely  proportioned  to  the 
radii ;  and  the  areas  of  the  circles  will  be  as  the  times  of  de- 
scribing  them,  or  the  times  will  be  directly  as  the  squares  of  the 
radii.  The  centrifugal  forcp  will  therefore  be,  §  64,  inversely  as 
the  cubes  of  the  radii,  or  distances  from  the  axis ;  it  may,  there- 
fore, in  approaching  the  vein,  give  the  particle  a  force  that  will 
enable  it  to  resist  the  action  of  the  descending  particles  of  the 


Book  Vl.~\  LIQVIDS.  407 

vein  that  it  tends  to  join.  The  greater  the  height  whence  the 
latter  have  descended,  the  greater  will  be  their  action  to  destroy 
the  centrifugal  force,  which  will  be  constant ;  and  hence,  the 
cavity  that  will  be  formed  by  the  latter,  on  the  surface  of  the  liquid, 
will  increase  as  the  depth  of  liquid  in  the  vessel  diminishes.  The 
figure  of  the  section  of  the  cavity  has  been  investigated  on  these 
principles  by  Venturi,  and  has  been  shown  to  be  a  curve  convex 
towards  the  axis  of  the  vein.  The  curve  is  one  of  the  third 
order. 


408  THE  DISCHARGE  [Book    VI. 


r-         CHAPTER  HI. 

OF  THE  DISCHARGE  OF  LIQUIDS  THROUGH  SHORT  PIPES  OR 
ADJUTAGES. 

403.  When  a  liquid,  instead  of  flowing  through  an  orifice  pla- 
ced in  a  thin  plate,  issues  by  a  short  pipe  ;  and  if  the  pipe  and 
the  liquid  be  of  such  materials  as  will  act  mutually  upon  each 
other  by  the  attraction  of  cohesion,  the  edges  of  the  orifice  will 
exert  a  force  upon  the  filaments  of  liquid  in  contact  with  them  ; 
the  consequence  of  this  should  be  their  deviation  from  their  ori- 
ginal direction  towards  the  vertical,  and  a  consequent  increase  in 
the  dimensions  of  the  vena  contracta. 

This  theory  is  fully  confirmed  by  experiment,  whence  it  appears, 
that  the  adaptation  of  pipes,  to  the  orifices  whence  liquids  issue, 
increases  the  quantities  discharged.  Such  additional  tubes  are 
called  Adjutages. 

404.  Adjutages  of  different  forms,  have  different  degrees  of  ad- 
vantage in  this  respect,  that  can  only  be  determined  by  experi- 
ment. 

When  a  cylindrical  tube  is  adapted  to  a  circular  orifice,  the  dis- 
charge is  increased,  until  the  length  amount  to  four  times  its  di- 
ameter; after  this  limit,  it  again  decreases  in  consequence  of  fric- 
tion in  the  tube. 

The  increase  in  the  discharge  is  in  the  ratio  of  82  :  62. 

The  same  increase  still  takes  place,  if  the  tube  be  contracted  in 
the  form  of  a  frustum  of  a  cone,  whose  altitude  is  at  the  distance 
of  half  the  radius  from  the  orifice,  and  whose  lesser  base  has  an  area 
of  T6^,  of  the  area  of  the  orifice ;  and  again  spread  out  to  its  ori- 
ginal size,  by  the  adaptation  of  another  conic  frustum. 

If  the  first  cone  be  merely  inserted  in  a  cylindric  tube,  the  in- 
crease is  only  in  the  ratio  of  77  :  62. 

If  the  tube  have  the  form  just  described  of  two  truncated  cones, 
adapted  to  each  other  at  their  lesser  bases  ;  the  greater  bases  ha- 
ving the  same  area  with  the  orifice,  the  lesser  that  of  the  vena  con- 
tracta, or  T6T2¥  ;  the  cone  next  the  orifice  an  altitude  equal  to  its 
radius,  the  other  cone  an  angle  at  the  vertex  of  36°,  the  discharge 
is  increased  in  the  ratio  of  92  :  62.  If  the  latter  cone  be  length- 
ened until  the  area  of  its  greater  base  becomes  one  half  more  than 
that  of  the  orifice,  the  discharge  is  increased  in  the  ratio  of 
94  :  62. 


Book  VI.]  OF  LIQUIDS.  409 

Thus  it  appears  that  in  an  adjutage,  a  part  of  which  is  contracted 
to  the  area  of  the  vena  contracta,  or  to  no  more  than  .62  of  the 
orifice,  the  actual  discharge  may  approach  within  .06  of  the  theo- 
retic. 

If  then  there  exists  a  right  to  draw  water  through  a  pipe  of 
given  dimensions,  the  quantity  determined  by  theory  may  be 
increased  in  the  ratio  of  132  :  100,  by  merely  uniting  the  pipe 
to  the  receiver  by  a  truncated  cone,  the  area  of  whose  lesser  base 
is  that  of  the  pipe;  whose  larger  base  has  an  area  that  bears  to 
theless,  the  ratio  of  100  :  62  ;  and  whose  altitude  is  half  the  diame- 
ter of  the  greater  base.  A  near  approach  is  obtained  to  this  form, 
by  making  the  diameters  of  the  base  as  10  :  8,  and  the  height  of 
the  cone  fths  of  the  diameter  of  the  tube. 

A  still  fartner  increase  in  the  ratio  of  150  :  100  may  be  ob- 
tained by  making  the  tube  spread  out  at  its  place  of  discharge,  in 
a  conical  form,  at  an  angle  of  1-6°  with  its  axis  ;  and  this  increase 
will  be  obtained,  even  if  a  cylindrical  tube  of  considerable  length 
intervene  between  the  two  cones. 

It  has  been  stated  by  some  writers  that  this  last  increase  wirl-. 
take  place,  whatever  be  the  length  of  the  intervening  cylindrical 
tube.     But  this  is  not  the  case  beyond  that  limit  at   which  the 
velocity  of  the  water  in  the  pipe  becomes  constant,  or  when  the 
retarding  and  accelerating  forces  counteract  each  other. 

Similar  results  take  place  in  channels  of  forms  other  than  cyl- 
indric;  in  them  all,  an  increase  of  the  liquid  they  will  carry,  may 
be  effected  by  giving  the  tubes  the  forms  the  liquid  would  assume 
under  the  mutual  action  of  its  particles. 

405.  These  results,  in  the  case  of  tubes  of  circular  section,  are 
very  remarkable,  and  are  worthy  of  exhibition  in  a  tabular  form. 

TABLE 

Of  the  quantities  of  a  liquid  discharged  in  equal  times  from  adjutages  of 
different  figures. 

From  an  orifice  in  a  thin  plate,        .         .         .         0.62 

Through  a  short  cylindrical  tube,  whose  area  is 

the  same  as  that  of  the  orifice,     .         .  0.82 

Through  a  tube  contracted  at  the  distance  of  half 

its  diameter  from  the. orifice  to  an  area  of  .62,         0.82 

Through  a  tube  of  the  figure  of  two  truncated 
cones,  whose  least  base  is  .62,  and  whose 
two  greater  bases  are  equal  to  the  orifice,  .  0.92 

Through  a  tube  formed  of  similar  cones,  whose 
length  is  increased  until  the  area  of  its  place 
of  discharge,  is  one  half  greater  than  that  of 

the  orifice, 0.94 

52  ^  N 


\ 


410  THE  DISCHARGE.  [Book    VL 

Theoretic  discharge,      .         .         .         .         .         1.00 
Discharge  through  a  given  aperture,  connected 
with  the  reservoir  by  a  truncated  cone  of 
which  the  aperture  is  the  lesser  base,     .         .         1.32 
Discharge  through  the  same  aperture,  connected 
with  the  reservoir  in  the  same  manner,  and 
which  has  another  truncated  tube  adapted  to 
it,  the  angle  of  whose  vertex  is  32°,     .      .  V  _•     1-50 


/ 


. 


Book   VI]  OF  LIQUIDS.  41  I 

^A  ten  Cf':-  rro  i  - 


CHAPTER  IV. 

OF  THE  MOTION  OP  WATER  IN  PIPES. 

406.  When  the  tube  that  is  adapted  to  an  orifice,  by  which 
water  flows  from  a  reservoir,  is  of  a  length  greater  than  four  times 
the  diameter  of  the  orifice,  the  velocity  is  retarded  ;  this  retarda- 
tion is  caused  by  a  resistance,  arising  from  the  friction  of  the 
liquid  against  the  sides  of  the  tube.  Under  the  action  of  this  re- 
sistance, the  velocity  of  the  liquid,  which  at  first  varies  with  the 
square  root  of  its  depth,  will  finally  become  constant. 

The  law  which  this  resistance  follows,  has  been  determined  by 
experiment.  It  has  been  thus  found  to  be  a  function  of  the  velo- 
city, and  of  such  a  nature  that  it  may  be  conveniently  divided 
into  two  parts  ;  the  first  of  which  is  directly  as  the  velocity  ;  the 
second  directly  as  its  square.  The  resistance  also  varies  with  the 
surface,  by  which  the  liquid  is  in  contact  with  the  channel  in 
which  it  runs. 

To  express  this  law  analytically  : 

Let  v  be  the  velocity  ; 

a,  and  /?,  constant  co-efficients,  determine)!  by  experiment  ; 

s  the  surface  ; 

The  friction,/,  will  be 

/=«(o*+j8««)  (417) 

When  a  liquid  moves  in  a  tube  of  uniform  diameter  with  a  con- 
stant velocity,  the  current  fills  the  whole  of  the  tube  ;  the  parti- 
cles of  the  liquid  in  immediate  contact  with  the  tube,  will  be  most 
resisted  by  friction  ;  but  in  consequence  of  the  viscidity  common 
to  all  liquids,  they  will  receive  motion  from  the  neighbouring  par- 
ticles, and  will  in  turn  retard  them  ;  hence,  although  the  velocity 
of  all  the  particles  situated  in  a  given  transverse  section  of  the 
tube  is  not  constant,  it  may,  without  any  sensible  error,  be  con- 
sidered as  such. 

Let  us  then  suppose  that  the  space  occupied  by  a  liquid,  that 
has  acquired  an  uniform  velocity  in  a  tube,  is  divided  into  a  great 
number  of  layers,  infinitely  thin,  by  means  of  planes  perpendicu- 
lar to  the  axis  of  the  tube.  Let  A,  B,  C,  D,  be  one  of  the  layers 

into  which  the  fluid  is  divided.  The 
motion  being  uniform,  the  resultant 
of  all  the  forces  that  act  upon  it  is 
=0,  or,  §  39,  they  are  in  equilibrio. 
Among  the  forces  that  accelerate 
are  the  fluid  pressures  ;  if  that  upon 
the  unit  of  surface  of  the  face  AB, 


MOTION  OF  [Book  VI. 

be  p,  that  on  the  unit  of  the  face,  C,  D,  will  be 

p+dp. 

The  forces  which  oppose  the  motion  of  the  liquid  will  be  : 
(1).  The  difference  of  the  fluid  pressures  on  the  opposite  sur- 
faces of  the  layer.     If  a  be  the  area  of  these  surfaces,  this  resis- 
tance will  be 

a  dp. 

(2).  The  friction.     This  as  has  been  seen,  (417)  will  be  rep- 
resented by 


but  in  a  thin  layer,  we  may  substitute  the  circumference,  c,  of  the 
pipe,  multiplied  by  the  differential  of  the  length  /,  for  s,  and  this 
resistance  will  become 

c(a.v+(3v2)dl.  (418) 

On  the  other  hand,  the  force  which  tends  to  move  the  liquid  in 
the  tube,  is  that  component  of  its  weight  which  lies  in  the  direction 
of  the  axis  of  the  tube. 

Let  i  be  the  inclination  of  this  axis  to  the  vertical  ;  the  whole 
weight  of  the  layer  will  be  • 

a  dig  ;  - 
and  its  component  in  the  direction  of  the  axis  of  the  tube, 

a  dig  cos.  «  . 
If  the  difference  of  level  of  the  points,  A  and  C,  be  dz,  we  have 

dz=dicos.i,  (419) 

therefore 

a  dig  cos.  i=agdz  .  (420) 

When  the  motion  is  constant,  §  39,  this  force  must  be  in  equi- 
librio  with  the  two  first,  or 

a  gdz=a  dp+c  (ar+/3r2)  dl  .  (421) 

Integrating,  and  introducing  for  the  constant  quantity,  the  initial 
pressure  at  the  origin  of  the  tube,  P,  we  have 

a  gz=a(p—P)+c(av+(3v2)  ;  (422) 

which,  when  /  becomes  equal  to  the  length  of  the  tube,  becomes, 
calling  the  pressure  at  its  place  of  discharge,  P', 

agz=a(Pf—  P)+c(a.v+(3v2)l  ;  (423) 

whence  we  obtain 


.          (424) 

If  the  diameter  of  the  tube  be  D, 


and 


P--/P' p\ 

ar+/3va=|  D  -»_  — '  .  (425) 


-*V.'v 

Book  VI.}  WATER  IN  PIPES.  413 

If  H  be  the  column  of  liquid  that  presses  on  the  origin  of  the 
tube,  and  H',  that  which  presses  on  its  place  of  discharge,  we 
have 

P=gH,  P'=£H'; 

and  substituting  we  obtain 

_  T37    I-TT 

.  (426) 


If  the  origin  of  the  ordinate,  z,  be  taken  at  the  surface  of  the  col- 
umn, H, 

z—  H'-fH=H—  H';  (427) 

and  if  the  discharge  take  place  in  the  open  air, 

z—  H'+H=H  ; 
in  which  case 

at>+/3i>2=£I)g3-.  (428) 

By  the  researches  of  Prony,  who  compared  fifty  different  ex- 
periments made  on  tubes,  for  conveying  water, 

-=0.00017, 
g 

fi  * 

/-=0.003416, 

S  .  ,  *  :$k 

therefore,  the  quantities  being  estimated  in  irietres, 

1  TT 

0.00017  v-f  0.003416  v2=-Vg.-j.-  (429) 

Whence  we  obtain,  by  taking  a  value  for  v,  deduced  in  a  particu- 
lar case  from  experiment, 

/  H\ 

0=_0.024S829-r-  V  (0.000619159+717.885  D2  -yj       (430) 

which  becomes,  when  the  quantities  are  estimated  in  English  feet, 

DHx 

0.02375+8201.6  —  )  .  (431) 

The  co-efficients  a  and  /3,  become,  if  applied  to  the  English 
foot  as  the  unit, 

a=0.00017, 
/3=0.000104; 
and  neglecting  the  term  that  involves  the  velocity  simply  ;  which, 


414  MOTION  op  [Book  VI. 

as  the  co-efficient  is  small,  may  be  done,  if  the  velocity  be  not 
great,  without  any  sensible  error,  we  have 

1  TT 

0.000104  7>3=-  D  p  (432) 

and 

H 

D9=2404 Dy  ;  (433) 

»=46.82  V  (D  y)  .  (434) 

If  the  diameter  of  the  tube  be  estimated  in  English  inches,  as 
is  most  usually  the  case,  and  the  other  quantities  in  feet, 

TT 

u3=200  D  y  ;  (435) 

and 

t>  =  14.142x/(Dy).  (436) 

If  Q  represent  the  quantity  discharged  by  the  tube  in  the*  unit 
of  time 


Q=*  -4-.  (437) 

and 

4Q 
t>=^p;  (438) 

substituting  this  in  the  equation  (428),  and  neglecting  the  first 
power  of  v,  we  have 

16  Q3     1      H 

and  if  we  make 

t_64/3 

6~  *•»   » 
we  have 

I 
and 


calculating  the  numeric  value,  we  obtain 

6=0.0000688, 
and 

Q=38.13>/(D5T),  (441) 

in  which  all  the  lineal  dimensions  are  in  English  feet,  and  the 
quantity  in  cubic  feet. 


Book  VL]  WATER  IN  PIPES.  415 

For  the  quantity  in  cubic  feet,  when  the  diameter  D  of  the 
tube  is  given  in  inches,  we  have  • , 

/       Wx 

(442) 

For  the  quantity  in  English  statute  gallons,  D  being  in  inches 
as  before, 

Q=  .  48635  N/  (D5  y .  (443) 

For  the  quantity  in  standard  liquid  gallons  of  the  Slate  of  New- 
York. 

TTv 

(444) 

The  formulas  for  the  value  of  D,  mfien  Q,  H,  and  /,  are  given, 
can  be  easily  obtained  from  the  foregoing,  the  fundamental  ex- 
pression being, 

i  J\. 

(445) 

The  formula  (434),  is  similar  to  that  of  Prony,  for  metres, 
which  is 

H\ 

T)  •  <446) 

but  which  reduced  to  English  measure,  would  be 

Hv 
T)'  (447) 

the  co-efficient  being  48.5,  instead  of  46.82,  as  we  have  made  it. 

Prony's  formula,  however,  appears  to  be  in  excess,  except 
when  the  velocity  is  considerable ;  that  of  (434)  is  probably  in 
defect,  except  at  small  velocities. 

The  formula  of  Du  Buat,  who  led  the  way  in  these  investiga- 
tions, is 

n    n      //I         /%    *  \  (448.) 


In  which, 

V  is  the  velocity  in  English  inches ; 
d  half  the  radius  of  the  pipe ; 

H 

*  the  mean  slope  of  the  pipe  which  is  equivalent  to  -y   of  Pro- 
ny's formula. 

log.  The  hyperbolic  logarithm  of  the  quantity  to  which  it  is 
prefixed. 

The  computation  by  this  formula  may  be  facilitated  by  means 
of  subsidiary  tables,  the  best  set  of  which  are  to  be  found  in  the 
Edinburgh  Cyclopedia,  article,  Hydrodynamics. 

:{.jr!'  m&* ;' W^^#$^  • 

•j  •     ^-M'^-^8'1 

•  4  •'.  .4.°    , 


. 

416  MOTION  OP  [Book  VI. 

The^  formula  of  Eytelwein,  is  in  English  feet. 
DH 
50 

407.  The  foregoing  investigation  is  only  applicable  to  the  case 
of  a  pipe  of  uniform  slope,  lying  in  the  same  vertical  plane,  and 
of  a  constant  section.     If  the  diameter  be  not  constant,  the  dis- 
charge will  obviously  be  due  to  the  area  of  its  least  section ;  but 
will  be  affected  by  the  same  causes  that  influence  the  discharge 
of  fluids  through  orifices  and  adjutages.     The  velocity  may  be 
calculated    as   above,  and   be   multiplied    by  the   smallest  area 
of  the  pipe.     The  amount  thus  obtained,  must  then  be  increased 
or  diminished  by  the  use  of  the  co-efficient,  which  may  be  ob- 
tained from  the  table  in  &  405,  according  to  the  nature  and  form 
of  the  contraction.  ^* 

The  quantity  D,  used  in  the  calculation  of  the  velocity,  must 
be  the  general  diameter  of  the  pipe,  for  the  friction  will  obviously 
be  principally  due  to  it,  or  nearly  so,  and  not  to  those  portions 
that  are  contracted. 

The  necessity  of  continuing  a  pipe  of  uniform  bore,  from  the 
place  where  it  receives  the  liquid  it  is  to  carry,  to  the  place  where 
it  is  to  discharge,  is  therefore  manifest;  but  at  its  two  extremi- 
ties it  should  have  conical  adjutages. 

408.  Pipes  that  convey  water,  are  liable  to  two  species  of  ob- 
struction, that  tend  to  diminish  the  effective  areas  of  their  section, 
and  thus  lessen  the  quantity  they  would  otherwise  discharge. 
All  spring  or  river  water  contains  gaseous  matter :  this  will  often 
escape  and  separate  itself  in  consequence  of  its  expansive  force; 
hence  lodgments  of  air  may  take  place  in^the  higher  parts  of  the 
pipe,  and  where  the  pipe,  after  having  risen,  is  bent,  and  again 
descends.     A  self-acting  apparatus  has  been  planned  to  permit 
the  escape  of  such  lodgments  of  air.     It  consists  of  a  valve  of  the 
form  of  a  sphere,  that  is  placed  in  a  vertical  cylinder,  adapted  to 
the  upper  bends  of  the  pipe  ;  in  the  cap  that  closes  this  cylinder, 
a  hole  is  cut  for  the  seat  of  the  valve,  and  is  carefully  ground  to 
the  figure  of  a  hollow  zone,  of  a  sphere  of  the  same  radius  as  the 
spherical  valve.      The  valve  is  made  of  metal,  and-is  hollow,  in 
order  that  it  may  be  light  enough  to  be  buoyant  in  water.    When 
the  pipe  runs  full  of  water,  the  pressure  of  the  liquid  keeps  the 
sphere  closely  applied  to  its  seat  ;  but  when  a  lodgment  of  air 
takes  place,  the  sphere  falls  ;  the  valve  is  therefore  opened,  and 
the  air  escapes;  the  water  which  follows  lifts  the  sphere,  and  ap- 
plies it  to  the  seat  that  has  just  been  described.     The  only  pre- 
caution in  using  this  is,  to  take  care  that  the  valve  seat  shall  be 
lower  than  the  level  of  the  water  in  the  reservoir  whence  the 
pipe  is  supplied. 


Book  VL~\  WATER  IN  PIPES.  417 

A  simple  stopcock,  that  is  occasionally  opened,  will  answer 
the  same  purpose,  but  is  not  self-acting. 

The  pipe  may  be  interrupted  at  the  places  where  the  air  is 
likely  to  lodge  and  the  water  discharged  into  a  basin,  whence  it 
is  again  drawn  by  the  prolongation  of  the  pipe.  In  this  case,  all 
the  advantage  derived  from  the  superior  height  of  the  water  in 
the  original  reservoir  is  lost.  This  may  be  obviated  by  raising 
the  pipe  in  this  place,  by  artificial  means,  to  the  height  due  to 
the  velocity  of  the  liquid,  which  will  be  as  much  less  than  the 
original  head,  as  is  due  to  the  friction. 

Such  an  arrangement  is  called  a  Souterazi. 

It  possesses,  when  applied  to  very  long  lines  of  pipes,  an  im- 
portant advantage ;  for  the  water  conveyed  in  them  becomes  vapid 
and  disagreeable,  but  will,  by  exposure  to  air,  recover  its  quali- 
ties. 

Deposits  of  earthy  matter  often  take  place  in  the  lower  angles 
of  a  pipe.  These  arise  from  substances  that  are  either  mechani- 
cally mixed,  or  held  in  solution  in  the  water.  These  deposits 
may  be  removed  by  throwing  a  cork,  to  which  a  string  is  attached, 
into  the  pipe.  If  the  space  left  in  the  pipe  be  sufficient  to  ad- 
mit its  passage,  it  will  carry  one  end  of  the  string  to  the  place  of 
discharge,  and  an  instrument  for  cleansing  the  pipe  adapted  to 
the  other  end,  may  be  drawn  through  the  pipe  by  means  of  it. 

Stopcocks  may  be  adapted  to  the  lower  angles  of  the  pipe,  and 
opened  at  proper  intervals ;  the  current  they  cause  will  carry 
with  it  any  earthy  matter  that  has  not  become  indurated. 

Short  tubes  may  be  placed  beneath  the  lower  angles,  communi- 
cating with  the  pipe,  by  means  of  a  smaller  vertical  pipe.  The  de- 
posit will  take  place  in  them,  instead  of  the  main  pipe,  and  they 
may  be  removed  as  often  as  necessary,  and  cleansed. 

409.  If  the  pipe  be  not  of  uniform  slope,  or  do  not  lie  wholly 
in  the  same  vertical  plane,  the  water  moving  in  it  will  experience 
a  resistance  at  the  angles.  The  amount  of  the  resistance  has  been 
ascertained  by  the  experiments  of  Du  Buat.  He  found  it  to  be 
proportioned  to  the  square  of  the  velocity,  and  to  the  square  of 
the  sine  of  the  deviation  of  the  tube  from  its  original  direction, 
to  the  number  of  bends  or  elbows  in  the  tube. 

For  a  single  angle,  therefore,  this  resistance  may  be  thus  ex- 
pressed : 

R=mv2  sin.Hi, 

and  for  any  number  of  elbows, 

R=-mu22,  sin.st.  (450) 


53 


.418  MOTION  OF  [Book  VI. 

The  co-efficient,  m,  as  determined  by  Du  Buat,  is  in  French 
inches,  :.-.-5.*i 

=0.000336, 


2998.5 
in  metres,  iV!*-« 

m=0.0123  ; 
and  in  English  feet, 

m=0.039. 
In  the  latter  case  the  formula  (450)  becomes 

R=0.0039t;22.(sin.2*)  .  (451) 

This  formula  ceases  to  be  true  when  the  angles  of  deviation 
exceed  30°. 

410.  The  general  formulae,  (428)  and  (440),  for  the  velocity 
and  quantity  discharged  by  a  pipe,  are  only  applicable  to  the  case 
of  a  single  pipe  of  uniform  bore  throughout,  or  where  there  are 
a  few  definite  contractions  in  the  course  of  such  a  pipe.  They 
are  not  adapted  to  the  circumstances  of  lateral  tubes  that  diverge 
from  a  main  pipe,  for  the  purpose  of  distributing  a  liquid  to  dif- 
ferent points,  as  is  frequently  necessary  in  the  supply  of  cities 
with  water.  Into  these  the  water  will  enter  with  a  velocity,  that 
is  due  to  its  pressure  upon  the  part  of  the  main  pipe,  to  which  the 
lateral  tube  is  adapted. 

Water,  as  may  be  inferred  from  the  preceding  investigations, 
moves  in  a  tube  in  consequence  of  the  pressure  upon  its  origin, 
and  the  weight  of  the  particles  in  the  descending  branches,  and 
is  resisted  by  the  friction  against  the  pipe,  and  the  weight  of  the 
particles  in  the  ascending  branches.  One  part  of  the  moving 
power  is,  therefore,  employed  in  generating  the  velocity  of  the 
liquid  ;  another  in  overcoming  friction  ;  while  the  third  is  ex- 
pended upon  the  resistance  of  the  columns  that  act  in  a  direction 
contrary  to  the  motion.  The  last  is  the  principal  element  that 
determines  the  pressure  on  the  tube.  The  main  tube  may  there- 
fore be  considered  as  a  reservoir  whence  the  lateral  pipe  is  with- 
drawn, and  all  the  circumstances  determined  upon  the  principles 
that  we  have  applied  to  the  main  tube,  and  its  reservoir.  The 
pressure  on  any  given  point  will,  theoretically  speaking,  be  due  to 
the  difference  between  the  actual  height,  and  that  due  to  the  velo- 
city of  the  liquid.  This  pressure  cannot  be  always  practically 
determined  with  an  accuracy  sufficient  for  the  purpose.  It  will 
be  seen,  however,  in  the  following  investigation,  that  the  dis- 
charge of  the  lateral  pipes  may  be  determined  by  means  of  ex- 
pressions, into  which  the  pressure  does  not  enter. 

Let  Q,  be  the  quantity  of  water  the  main  pipe  would  deliver 
at  the  point  whence  the  first  lateral  pipe  diverges  ; 


Book  Vl.~\ 


WATER  IN  PIPES. 


419 


D,  the  diameter  of  this  pipe  ; 

L,  its  whole  length  ; 

X  X'  X"  ...  X"-1,  the  partial  lengths  of  the  main  pipe,  whose 
sum  is  equal  to  L  ; 

Z,  the  difference  of  level  between  the  water  in  the  reservoir, 
and  the  opening  of  the  first  lateral  tube  ; 

Z'  Z"  .  .  .  .  Zn~S  the  difference  of  level  between  each  two  con- 
secutive branches. 

H'  H"  H'" ....  H",  the  heights  due  to  the  pressures  on  the 
points  whence  the  successive  branches  diverge  ; 

q'  d'  I'  z'    ) 

q"  d"  I"  z"   \  similar  elements  for  each  separate  branch ; 

qn  fa  ln  zn    j 

cy  the  constant  quantity  in  the  formulae  (440),  to  (444),  accor- 
ding to  the  measure  employed ;  we  have  for  the  several  parts  of 
the  main  pipe,  and  its  branches  from  (440), 

w 


(6) 


(452) 


r7"  _  W" 


The  equations  being  in  all  to  the  number  of  2w,  or  twice  as 
many  as  there  are  lateral  pipes, 

To  eliminate  H',  H",  &c.,  we  first  combine  the  equations  (a] 

and  (c),  (a),  (c),  and  (e),  &c. ;  we  thus  obtain 

Q2X+(Q— q'}  2X'=c2DXZ+Z'— H")  ,  (g) 

Q2X+(Q— f/)  2X' +(Q— q'  ....  g«-7X»-'  = 

c2D5(Z+Z'- Z"-1—  Hn)  .  (h) 

Then  by  combining  the  equations  (a]  and  (6),  (d}  and  (g*),  &c. 
we  have 

Z— z'}  ,  (») 


(Jfc) 


[Q2X+(Q—  q'}2^  ____  +(Q—  q'  -  -  —  gn 
c2D5dn5(Z+Z'  .....  +Z"-'— 


420  LIQUIDS  IN  [Book  VI. 

From  the  last  equation  we  may  obtain  the  values  d'  d",  to 
which  are 

»5  -^ 

v 

(453) 


cD5(Z-r-Z"— a")— (Q2+(Q— q'W 
These  two  equations  are  sufficient  to  determine  the  law ;  and  in 
order  that  they  shall  give  rational  values  for  the  diameters  of  the 
lateral  pipes,  it  is  necessary  that  the  denominators  of  the  fractions 
should  be  positive*. 

"See,  "Essai  sur  les  Moytns  de  conduire  deleter  et  dc  distribuer  les  Eauz,  par  M. 
Genieys. 


Book   VL\  OPEN  CHANNELS.  421 

V 

• 

CHAPTER  V. 

OF  THE  MOTION  OF  LIQUIDS  IN  OPEN  CHANNELS. 

411.  When  a  liquid  issuing  from  a  reservoir  enters  into  an 
open  channel,  the  general  direction  of  the  bed  must  be  inclined 
downwards,  otherwise  it  would  not  continue  to  flow ;  and  the 
surface  will  have  a  slope  from  the  reservoir  towards  the  place  of 
discharge.  Being  acted  upon  by  the  force  of  gravity,  the  liquid 
will  have  a  tendency  to  assume,  an  accelerated  velocity.  It  rarely 
however  happens,  and  only  when  the  slope  is  very  great,  or  the 
length  of  the  channel  small,  that  this  acceleration  does  occur;  in 
some  cases  the  velocity,  so  far  from  increasing  with  the  distance 
from  the  reservoir,  or  source,  diminishes.  In  most  instances  the 
mean  velocity  is  found  for  long  distances,  to  be  uniform,  and  to 
change  only  with  changes  in  the  nature  and  character  of  the  bed. 

When  a  stream  flows  in  a  channel  with  uniform  mean  velocity, 
it  is  said  to  be  in  train ;  this  can  only  occur  when  no  accelerating 
force  acts,  or  when  the  sum  of  the  accelerating  and  retarding 
causes  is  =0. 

The  circumstances,  then,  of  the  uniform  motion  of  a  liquid  in 
an  open  channel  of  uniform  section,  may  be  made  the  basis  of  the 
theory  of  the  motion  of  liquids  in  open  channels  of  any  figure  or 
variety  of  dimension  whatsoever;  and  the  variations  from  the 
simple  theory  which  the  change  of  dimension  may  produce,  can, 
if  necessary,  be  applied  as  corrections  to  the  inferences. 

The  principal  retarding  force,  to  which  the  motion  of  liquids 
in  open  channels  is  liable,  is  the  friction  upon  their  beds.  This, 
according  to  the  experiments  of  Coulomb,  will  be  a  function  of  the 
velocity  and  of  the  surface  directly,  and  be  inversely  as  the  area 
of  the  section  of  the  stream ;  and  as  in  the  case  of  tubes,  that  part  of 
the  resistance  that  is  a  function  of  the  velocity,  has  two  terms ;  in 
one  of  which  the  first,  and  in  the  other,  the  second  power  of  the 
velocity  are  involved.  This  observation  forms  the  basis  of  the 
theory. 

This  retarding  force  may  be  thus  expressed : 

f=g-(av+(3v*).  (454a) 

w  '  *',  - 

In  which  formula  a  and  /3  are  constant  co-efficients,  determined 
in  some  particular  case  by  experiment ; 
/,  the  friction ; 
of,  the  measure  of  the  gravitating  force  ; 


422  LIQUIDS  IN  [Book  VI. 

a,  the  length  of  the  perimeter  of  that  surface,  which  is  in  contact 
with  the  liquid ; 

to,  the  area  of  a  transverse  section. 

Now  let 

1=  the  length  of  the  axis  of  the  channel ; 

1  H=  the  difference  of  level  between  the  places  where  the  chan- 
nel receives  and  delivers  the  liquid  ;  we  may  obtain  by  a  course 
of  reasoning  similar  to  that  in  §  402. 

to  H 


The  quantity  -  ,  is  that  which  is  called  the  Hydraulic  mean 
s 

depth,  or  Radius ;  this  is  usually  represented  by  R.     The  quan- 
tity — ,  is  the  slope  or  inclination  of  the  tube,  which  is  represented 

by  I.     Using  these  symbols,  we  have 

av+f3v2=gRI.  (454) 

According  to  the  investigations  of  Prony,  the  constant  numbers 
applicable  to  this  equation,  are,  after  division  by  g,  when  the  mea- 
sures are  taken  in  metres, 

a=0.0004445, 
/3=0.0003093; 
or  in  English  feet, 

a=0.0004445, 
/3=0.0000912; 
and  neglecting  at%  we  have 

.0000912t>2=RI ;  (455) 

whence 

t>2=12000  RI, 
t?=109.53v/RI. 

Eytelwein  makes  the  numbers  for  English  feet, 
a=0.000243, 
^=0.00001113  ; 
whence  we  obtain,  again  neglecting  m«, 

v=94.87VRI  .  (456) 

If  we  employ  the  same  constant  number  for  /3,  that  we  have 
made  use  of  in  tubes,  §  402,  we  have 

v  =  93.64v/RI;  (457) 

or  taking  the  simpler  formula  of  Prony,  as  in  (447), 

VZ-97VRI  .  (458) 

By  the  application  of  experiment  to  determine  the  value  of  u, 

in  a  particnlar  case,  Prony  obtains  a  general  formula,  applicable 


Book    F/.]  OPEN    CHANNELS.  423 

both  to  tubes  and  open  channels ;  this  is  as  follows,  the  measures 
being  in  English  feet : 

-y=_0.154+N/  (0.0238-1-32806  G)  .  (459) 

In  this  formula,  in  the  case  of  pipes, 

G^ID^-;  (460) 

in  the  case  of  open  channels, 

G=RI=^.^.  '.-,-    (461) 

In  the  former  case,  therefore,  it  is  identical  with  (431). 

412.  The  velocity,  v,  in  the  foregoing  investigations  is  the 
mean  velocity,  and  will  not  be  that  of  all  parts  of  the  stream. 
Those  portions  which  are  nearest  to  the  sides  and  bottom  of  the 
channel,  will  be  directly  resisted  by  them  ;  and  although  in  con- 
sequence of  the  xviscidity,  they  will  derive  motion  from  those 
more  distant,  and  will  retard  them,  the  latter  will  have  the 
greater  velocity.  On  the  other  hand,  the  liquid  pressure  will 
tend  to  accelerate  the  threads  of  fluids  which  lie  deepest  beneath 
the  surface  of  the  stream.  By  the  combination  of  these  two  ac- 
tions, the  greatest  velocity  in  a  symmetric  open  channel  will  be 
in  its  middle,  and  at  a  small  distance  below  the  surface.  Between 
the  greatest  velocity  at  or  near  the  surface,  and  the  mean  velocity, 
there  is  a  constant  relation,  as  has  been  shown  by  the  experimen- 
tal researches  of  Du  Buat.  The  formula  derived  by  him  from  his 
experiments  is  however  faulty,  and  has  been  converted  by  Prony, 
into  the  following  form  : 

Let  Vj  be  the  mean  velocity ; 
V,  the  greatest  velocity ; 

-*&•• 

From  this,  may  be  derived  the  following  mean  value  of  v, 

v =0.816458  V;  (463) 

or,  what  is  in  most  cases  sufficiently  accurate, 

0=4  V  '  (464) 

o 

413.  When  the  channels  in  which  streams  run,  are  neither  of 
uniform  section  throughout,  nor  directed  in  a  straight  line,  varia- 
tions from  the  above  theoretic  inferences  take  place. 

Bends  and  elbows  in  the  stream  obstruct  its  course,  and  dimi- 
nish the  discharge ;  the  current  will  in  such  cases  be  directed  from 
one  of  the  points  towards  the  next,  which  it  will  tend  to  wear 
away,  and  in  the  bays  that  intervene,  eddies  take  place. 

The  formation  of  eddies  may  be  thus  explained.      In  conse- 


424  MOTION  OP  LIQUIDS.  [Book   VI. 

quence  of  the  viscidity  of  liquids,  a  current  that  is  in  motion, 
tends  to  carry  with  it  the  neighbouring  masses  of  a  similar  fluid 
nature  ;  thus,  if  a  current  be  passing  through  a  space  greater  than 
it  is  capable  of  occupying,  without  changing  its  velocity,  the 
lateral  fluid  will  join  the  stream,  and  tend  to  increase  its  quan- 
tity;  by  this  flow  the  level  will  be  lowered,  and  the  pressure  of 
the  adjacent  masses  will  impel  a  current  towards  the  lowest  point, 
which  will  be  that  whence  the  fluid  is  first  drawn.  In  a  wide 
channel,  the  main  stream  will  be  increased  by  this  action,  and  on 
one  or  both  sides  of  it,  a  current  will  be  formed  in  an  opposite 
direction,  into  which,  when  the  space  is  again  contracted,  or 
another  point  reached,  the  excess  of  fluid  that  has  united  itself 
with  the  main  stream  will  flow,  and  thus  keep  up  the  circulation. 
If,  on  the  other  hand,  the  increase  in  the  section  of  the  channel  be 
permanent,  eddies  will  at  first  be  formed,  but  the  velocity  of  the 
main  stream  will  gradually  diminish,  until  it  become  capable  of 
filling  its  new  bed. 


Book  VI.  ]  or  RIVERS.  425 


CHAPTER  VI. 
OF  RIVERS. 

414.  The  vapour  which  is  raised  from  the  whole  surface  of  the 
earth,  and  particularly  from  the  ocean,  tends  to  distribute  itself 
uniformly  according  to  the  mechanical  law,  mentioned  in  §  355, 
and  known  by  the  name  of  its  discoverer,  Dalton.  Thus,  the  ex- 
cess of  moisture  furnished  to  the  atmosphere,  from  the  surface  of 
the  sea,  is  borne  towards  the  land  ;  upon  the  latter,  as  will  here- 
after be  shown,  the  causes  that  produce  precipitation  are,  in  gene- 
ral, more  frequent  than  on  the  ocean  ;  hence,  more  moisture  falls 
on  the  surface  of  the  land,  than  is  evaporated  from  it.     The  loss 
arising  from  this  excess  of  precipitation  is  again  supplied  from  the 
ocean,  in  conformity  with  the  law  we  have  cited.     It  thus  hap- 
pens that  in  most  parts  of  the  continents  and  islands,  the  quantity 
of  moisture  that  falls  to  the  surface,  in  the  form  of  rain,  hail, 
snow  and  dew,  exceeds  that  which  is  evaporated  from  their  sur- 
face. This  excess  partly  runs  over  the  surface  of  the  ground,  and 
partly  penetrates  into  it.     In  the  latter  case  it  frequently  meets 
impervious  strata  ;  along  these  the  water  is  carried  by  its  gravity, 
until  they  break  out  to  the  day.  The  water  will  there  form  springs. 
These  unite  with  that  running  upon  the  surface,  and  descend  to 
the  lowest  points  of  vallies,  where  they  form  lakes,  or,  if  the 
slope  be  sufficient,  streams  of  various  magnitudes.     Lakes  when 
they  increase  to  such  a  height  as  to  overtop  the  lower  parts  of 
their  barriers,  or  acquire  a  sufficient  pressure  to  force  their  way 
through  them,  also  give  rise  to  streams.     Such  streams  running 
in  the  lower  parts  of  vallies,  and  upon  the  lines  of  the  greatest  slope, 
unite  at  the  junction  of  two  or  more  vallies,  and  mix  their  waters 
in  a  channel  of  greater  magnitude  ;  and  thus,  by  successive  junc- 
tions, if  the  extent  of  country  be  sufficient,  rivers  of  great  mag- 
nitude are  formed. 

415.  The  formation  of  the  beds  and  channels  of  rivers  appears, 
even  on  the  most  cursory  observation,  to  have  been  effected  by 
the  action  of  their  own  waters  :  from  the  continued  effort  of  their 
streams,  a  species  of  equilibrium  has  taken  place  between  the  mo- 
tion of  their  currents,  and  the  resistance  of  the  soil  over  which  they 
run.     Thus  :  when  the  velocity  is  very  great,  the  beds  are  com- 
posed of  solid  rock;  in  the  next  stage  of  diminishing  velocity, 
the  beds  are  composed  of  large  rolled  stones  ;  when  the  velocity 
decreases  still  farther,  the  bottom  is  composed  of  gravel ;  then  of 

54 


426  or  RITERS.  [Book  VL 

sharp  sand ;  and  finally,  when  the  water  becomes  stagnant,  of 
fine  argillaceous  particles,  or  mud. 

Still  the  equilibrium  between  the  active  force  of  the  stream 
and  the  resistance  of  its  bed,  is  not  absolutely  perfect  ;  a  slow 
and  gradual  action  takes  place  on  all  parts  of  its  bed,  impercepti- 
ble, except  in  the  larger  class  of  streams,  or  under  extraordinary 
circumstances ;  but  this  action  is  sure,  and  after  the  lapse  of 
years  will  be  found  to  have  produced  most  important  effects. 

It  has  been  determined  by  the  observations  of  Du  Buat,  that 
fine  clay  will  not  resist  the  action  of  a  current,  whose  velocity, 
at  bottom,  exceeds  three  inches  per  second  :  fine  sand  begins  to 
resist 'when  the  velocity  falls  below  six  inches  ;  coarse  sand,  at 
a  velocity  of  eight  inches;  gravel,  at  velocities  from  seven  to 
tw.elve  inches  ;  pebbles  of  an  inch  in  diameter,  at  two  feet ;  and 
angular  fragments,  of  the  size  of  an  egg,  at  three  feet  per  second. 

Rivers  cannot  be  considered  as  forming,  throughout  their  whole 
course,  channels  of  uniform  slope  ;  on  the  contrary,  they  are,  as 
a  general  rule,  most  inclined  near  their  sources,  and  become  less 
and  less  so  as  they  approach  the  sea.  They  may,  however,  in 
most  cases,  be  divided  into  portions;  each  of  these  may  be  con- 
sidered as  composed  of  a  current  flowing  with  a  constant  area, 
and  having  an  uniform  slope  at  its  surface.  These  portions  are 
frequently  separated  by  marked  physical  barriers. 

Under  the  long  conflict  of  the  active  forces  of  running  water, 
and  the  passive  resistance  of  their  beds,  the  latter  have,  in  almost 
all  cases,  assumed  a  state  of  apparent  permanence ;  the  changes 
that  take  place  in  them  are  slow,  and  only  perceptible  by  the 
comparison  of  their  states  at  long  distant  periods. 

416.  Rivers  are  subject  to  an  increase  and  diminution  in  the 
quantity  of  their  waters.  This  variation  sometimes  bears  but  a 
small  proportion  to  their  mean  magnitude,  at  others  is  of  great 
extent.  It  is  in  some  cases  periodic,  varying  with  the  season, 
being  produced  by  the  melting  of  the  snows,  in  other  cases,  is 
subject  to  no  fixed  law. 

When  the  variation  in  the  bulk  of  a  stream  is  small,  its  bed 
has  usually  for  its  section  a  concave  curve,  whose  versed  sine 
bears  a  greater  ratio  to  its  chord,  when  formed  in  earth  that  op- 
poses a  great  resistance,  and  of  course,  when  the  stream  is  rapid, 
than  it  does  in  soils  of  less  tenacity.  The  resistance  of  solid  rock 
may  render  this  rule  untrue  when  the  bed  is  formed  in  that  sub- 
stance. ;«.••:•• 

When  the  variation  in  the  bulk  of  the  stream  is  great,  and  its 
slope  small,  there  is  usually  a  bed  of  the  same  form- as  in  the  other 
case,  suited  to  convey  the  stream  when  it  does  not  much  exceed 
its  mean  magnitude  ;  this  is  enclosed  on  one  or  both  sides  by  the 


Book  VI.]  OF  RIVERS.  427 

alluvial  deposits  of  the  river.  These  usually  assume  the  shape 
of  an  inclined  plane,  sloping  from  the  bank  of  the  main  stream 
towards  the  adjacent  country.  The  manner  in  which  this  shape 
is  produced,  may  still  be  witnessed  in  the  streams  of  many  parts 
of  our  own  country.  The  river,  as  its  volume  increases,  flows 
with  greater  rapidity,  and  carries  with  it  an  increased  quantity  of 
earthy  matter  ;  when  it  rises  so  far  as  to  overtop  the  banks  which 
bound  it  when  not  swollen,  if  trees  and  bushes  grow  upon  them, 
they  will  catch  and  retain  the  larger  and  heavier  parts,  while  the 
lighter  alone  will  remain  suspended.  Thus  the  greater  propor- 
tion of  the  deposit  takes  place  on  the  edge  of  the  ordinary  bed. 
This  process  may  still  be  annually  witnessed  on  the  Mississippi ; 
and  the  height  of  the  natural  dyke  that  is  thus  formed,  is  in- 
creased by  the  quantity  of  drift  wood  that  is  retained  by  the 
nearest  obstacles.  This  natural  barrier,  so  long  as  the  periodic 
overflow  is  unimpeded,  more  than  counteracts  any  rise  that  may 
take  place  in  the  bed,  from  deposits  at  those  times  when  the 
stream  has  less  than  its  mean  velocity. 

It  frequently  however  happens  in  cultivated  countries,  that  it 
becomes  necessary  to  restrain  the  periodic  overflow.  For  this 
purpose  dykes  are  erected  on  the  borders  of  the  usual  channel, 
their  erection  being  facilitated  by  the  natural  form  of  the  adja- 
cent ground.  In  this  case,  the  substances  carried  by  the  stream 
are  deposited  in  its  bed,  instead  of  being  spread  over  the  whole 
surface,  to  which  the  inundation  had  before  extended.  The  level 
of  the  surface  of  the  stream  will  in  consequence  rise,  until  the 
slope  may  finally  become  too  small  to  convey  it,  and  it  may  seek 
an  outlet  in  other  directions.  It  is  thus  rendered  indispensable 
to  raise  the  dykes  to  correspond  to  the  increased  height  of  the 
surface  of  the  stream.  By  a  long  continued  process  of  this  kind, 
the  bottom  of  the  bed  of  the  Po  has  been  raised  higher  than  the 
level  of  the  adjacent  country,  and  the  surface  of  its  stream  over- 
tops the  houses  of  Ravenna  ;  so  also  the  ancient  channel  of  the 
Rhine  has  been  filled  up,  until  it  will  no  longer  carry  its  waters, 
and  they  have  sought  outlets  in  other  directions. 

A  similar  action  is  going  on  upon  the  main  outlet  of  the  Mis- 
sissippi, where  the  superior  magnitude  of  the  stream  makes  a 
change  in  its  bed,  and  in  the  height  of  its  waters,  when  full,  more 
marked  in  a  few^  years  than  it  is  in  the  Po  and  Rhine  in  cen- 
turies. 

A  river  that  requires  to  be  confined  by  dykes,  is  often  crooked 
in  its  course.  The  danger  of  overflow  may,  in  this  case,  He  les- 
sened upon  principles  derived  from  our  investigations.  It  will 
be  seen  by  reference  to  the  formula  (456),  that  the  velocity  varies 
with  the  square  root  of  the  slope  ;  if  then  the  course  be  rendered 


4*8  OF  RIVERS.  [Book  VI. 

straight,  the  fall  between  two  given  points  remaining  constant, 
while  the  distance  in  the  direction  of  the  stream  is  lessened,  the 
slope  is  augmented,  and  a  bed  of  given  dimensions  will  carry  a 
greater  quantity  of  water,  in  a  given  time. 

417.  Among  the  many  important  purposes  that  rivers  subserve, 
that  of  navigation  is  worthy  of  particular  notice. 

Rivers,  according  to  their  size  and  importance,  are  sometrmes 
navigable  for  vessels  of  various  descriptions  and  sizes ;  bqt  are, 
at  others,  frequently  liable  to  obstructions  and  impediments,  that 
either  interrupt  or  prevent  their  navigation  altogether. 

These  obstructions  and  impediments  may  be  arranged  in  seve- 
ral distinct  classes. 

(1.)  Rivers  maybe  obstructed  by  Falls  or  Cataracts,  consisting 
in  a  sudden  change  of  level. 

(2.)  Rocky  barriers  may  cross  the  bed  of  a  river ;  these  will 
prevent  its  forming  a  channel  sufficiently  large  to  discharge  its 
waters  with  its  usual  mean  velocity.  In  this  case,  the  flow  of 
the  higher  part  being  impeded,  the  water  accumulates  above  the 
barrier,  until  the  slope  becomes  so  great  immediately  over  it,  as 
to  increase  the  velocity  there  to  an  extent  adapted  to  the  discharge 
of  the  stream,  through  the  contracted  space.  Such  obstructions 
are  called  Rapids. 

In  large  streams  the  rocky  barrier  sometimes  lies  so  deep  that 
navigation  is  not  obstructed,  but  merely  embarrassed,  by  a  cur- 
rent of  increased  velocity.  Such  is  the  rapid  at  the  outlet  of 
Lake  Erie,  and  a  more  magnificent  instance  is  to  be  seen  in  the 
Race  in  Long  Island  Sound. 

In  small  streams  they  interrupt  the  navigation  altogether. 

(3.)  When  rivers,  after  running  in  a  mountainous  country, 
reach  one  of  less  slope,  their  velocity  is  diminished ;  the  earthy 
particles  they  had  before  been  able  to  carry  with  them,  are  in 
consequence  deposited,  'the  bed  is  filled  up,  and  the  river  seeks  a 
discharge  by  spreading  itself.  Thus  the  breadth  of  the  channel 
is  increased,  and  its  depth  diminished.  A  similar  consequence 
may  follow  when  the  velocity  of  a  river  is  lessened  by  its  meet- 
ing the  tide  within  its  own  channel.  By  a  combination  of  these 
two  causes,  the  obstructions  that  exist  in  the  Hudson  River,  near 
Albany,  have  been  formed. 

(4.)  Where  a  river  that  carries  large  quantities  of  earthy  mat- 
ter enters  the  ocean,  the  conflicting  action  of  the  two  masses  of 
water,  causes  a  cessation  of  motion  that  influences  a  deposit ;  bars 
are  thus  created  in  the  sea,  in  advance  of  the  mouth  of  the  river. 
In  the  case  of  large  rivers,  these  may  become  the  basis  of  islands, 
which  are  gradually  connected  with  the  main  land,  while  the  force 
of  the  current  sweeps  out  the  intervening  channels,  and  new  bars 


Book  VI.}  OF  RIVERS.  429 

are  formed  beyond  them.  In  this  manner  the  Deltas  of  rivers 
were  originally  deposited,  and,  in  some  instances,  still  continue 
to  protrude  themselves  into  the  sea. 

(5.)  A  river  may  carry  a  sufficient  quantity  of  water  to  admit 
of  navigation,  and  may  be  unobstructed  by  falls,  rapids,  or  shal- 
lows, but  may  be  so  rapid  as  either  to  prevent  an  ascending  trade, 
or  to  diminish  the  area  of  the  stream  so  far  as  not  to  admit  of  a 
vessel  floating  in  it. 

418.  Each  of  these  obstructions  has  its  appropriate  remedy, 
the  application  of  which  may  however  be,  in  some  cases,  imprac- 
ticable, either  from  physical  circumstances,  or  the  great  expen- 
diture they  involve. 

(1.)  When  a  river  is  obstructed  by  falls,  we  make  a  lateral 
channel,  fed  from  the  upper  level,  and  apply  locks  to  it,  upon 
principles  that  will  be  explained  under  the  head  of  canals. 

(2.)  If  rapids  have  upon  them  a  sufficient  depth  of  water  to  float 
the  vessels,  artificial  mechanical  means  may  be  brought  in  aid — 
thus  :  the  power  of  men  or  of  animals  may  be  applied  from  the 
shore  ;  powerful  steamboats  may  be  used  in  towing  ;  or  the  ves- 
sel may  have  wheels  adapted  to  it,  the  area  of  whose  paddles  is 
greater  than  its  own  section.  In  this  last  case,  if  a  barrel  be 
adapted  to  the  wheels,  and  a  rope  passed  around  it  and  fastened  to 
the  shore  above  the  rapid,  the  wheels  being  more  powerfully  acted 
upon  by  the  current  than  the  vessel  is,  will  turn  around,  coil  up 
the  rope,  and  drag  the  vessel  forward.  It  however  generally 
happens  that  the  impediment  of  a  rapid  must  be  overcome  by  a 
lateral  channel,  as  described  in  the  preceding  instance. 

(3.)  As  the  formation  of  shallows  arises  from  a  diminution  in 
the  velocity  of  a  stream,  it  may  be  prevented,  and  the  obstacle 
may  even  be  removed,  by  increasing  the  velocity.  This  may  be 
done  by  contracting  the  horizontal  dimensions  of  the  bed.  The 
upper  portions  of  the  stream  being  thus  retarded,  the  level  rises, 
until  the  slope,  as  in  the  natural  formation  of  a  rapid,  becomes 
sufficient  for  the  discharge.  With  this  increased  velocity,  the 
stream  will  have  sufficient  force  to  carry  away  the  earthy  matter 
it  before  deposited,  and  a  permanent  improvement  in  its  depth 
will  take  place.  \ 

As  a  stream  tends  to  continue  of  uniform  section,  even  in  an 
increased  channel,  forming  eddies  in  the  bays  and  hollows,  it  is 
not  necessary  that  this  contraction  should  be  effected  by  conti- 
nuous and  parallel  lateral  dykes.  It  is  sufficient  that  piers  be 
built  out,  alternately  from  each  bank,  at  distances  from  each  other 
equal  to  about  the  breadth  it  is  proposed  to  give  the  stream.  The 
heads  of  these  piers  should  be  arranged,  if  possible,  in  two  pa- 
rallel straight  lines,  in  order  that  the  stream  may  assume  a  straight 


430  OF  RIVERS.  [Book  VI. 

course,  in  which,  as  has  been  already  explained,  it  will  have  the 
greatest  velocity.  The  deposit,  which  was  before  uniformly  dis- 
tributed over  the  bed,  will  now  take  place  between  the  piers. 
In  all  such  cases,  a  straightening  of  the  channel  is  advantageous. 

If  islands  be  formed  in  the  shallow  parts,  as  often  occurs,  all 
the  branches  of  the  river  that  surround  them,  except  one,  should 
be  closed,  by  weirs  that  rise  to  the  ordinary  level  of  the  stream, 
and  which  will  therefore  permit  a  discharge  through  these  lateral 
channels  when  it  is  swollen.  If  there  be  several  islands  on  either 
side  of  the  main  channel,  it  may  be  often  advantageous  to  unite 
them  by  a  longitudinal  dyke,  in  order  to  prevent  the  stream  from 
spreading  between  them. 

(4.)  A  similar  principle  will  direct  the  operations  intended  for 
the  removal  of  bars  at  the  mouths  of  rivers.  It  becomes  neces- 
sary to  give  the  stream  such  a  velocity  as  will  make  its  force 
preponderate  over  the  resistance  of  the  ocean.  In  large  rivers, 
the  extent  of  the  bars  puts  them  beyond  the  reach  of  improve- 
ment. In  small  streams,  piers  may  be  built  out  from  the  main 
land,  from  each  side  of  the  mouth  of  the  river,  gradually  inclin- 
ing to  each  other.  The  waters  being  all  confined  between  them, 
and  acquiring  from  the  contraction  an  increased  velocity,  will 
not  only  cease  to  make  farther  deposits  upon  the  bar,  but  will 
carry. away  so  much  of  it  as  lies  between  the  piers.  The  deposit 
will  now  take  place  behind  the  piers  on  either  hand.  Where 
the  direction  of  the  river  makes  a  small  angle  with  the  line  of 
the  coast,  a  single  pier  or  jetty  may  be  sufficient.  Of  this  me- 
thod we  have  fine  instances  in  the  ancient  port  of  Dunkirk,  and 
in  the  present  port  of  Havre  :  and  similar  principles  have  been 
successfully  applied  at  Buffalo,  on  Lake  Erie. 

(5.)  A  rapid  stream  may  be  rendered  navigable  by  building 
dams  or  weirs  across  it  from  place  to  place :  between  each  two 
of  these,  the  depth  of  the  water  will  be  increased  and  its  velocity 
Diminished;  by  both  of  these  changes  navigation  may  be  ren- 
dered more  easy.  The  passage  of  these  dams  by  vessels,  was 
originally  effected  in  one  of  two  different  methods  ;  these  require 
illustration,  in  consequence  of  their  being  the  germs  of  more 
perfect  means,  that  are  still  in  use.  The  first  of  these  methods 
was  the  Sluice;  the  second,  the  Inclined  Plane. 

To  form  a  sluice,  an  aperture  is  left  in  the  mass  of  each  of  the 
dams  that  divide  the  river  into  successive  ponds,  of  different 
levels.  This  is  for  the  greater  part  of  the  time  closed  by  a  gate ; 
and  as  this  gate  will  have  an  unequal  pressure  on  its  opposite 
sides,  in  consequence  of  the  difference  of  level,  it  can  only  be 
opened  by  a  vertical  motion. 

In  the  canals  of  China,  the  gate  is  formed  of  heavy  beams, 
dropped  into  vertical  grooves,  made  in  the  masonry  of  the  dam. 


Book  VL~\  or  RIVERS.  431 

These  being  piled  upon  each  other,  the  lower  ones  are  thus  loaded 
with  a  weight  that  sinks  them  to  the  bottom  of  the  passage,  and 
they  oppose  an  effectual  barrier  to  the  flow  of  the  water  through 
them  ;  for  the  small  quantity  that  may  penetrate  between  their 
joints,  will  be  wholly  unimportant. 

When  the  gate  is  opened,  the  water  will  discharge  iteslf  with 
great  velocity  through  the  passage.  The  vessels  of  the  descending 
trade  are  committed  to  this  current:  and  as  they  would  run  the 
risk  of  being  swept  into  the  eddies  it  will  form,  men  are  stationed 
on  each  bank,  furnished  with  poles,  to  keep  them  in  the  direc- 
tion of  the  effluent  stream.  By  this  discharge,  the  level  of  the 
water  in  the  upper  pond,  is  lowered  ;  the  velocity  of  the  current 
that  flows  through  the  passage  is  diminished  in  consequence. 
Natural  agents,  such  as  the  force  of  men  or  animals,  are  then  ap- 
plied to  draw  up  the  vessels  that  carry  the  ascending  trade. 

If  the  change  of  level  be  considerable,  the  danger  in  passing 
down,  and  the  force  required  to  draw  the  vessels  up,  are  both  too 
great.  This  method,  therefore,  becomes  impracticable.  In  the 
infancy  of  inland  navigation,  the  inclined  plane  of  a  rude  and 
imperfect  form,  was  introduced  in  cases  of  this  sort.  In  a  part 
of  the  weir  was  introduced  a  mound  of  masonry,  of  the  form  of 
a  triangular  prism.  Of  the  two  inclined  planes  that  composed 
its  upper  surface,  one  descended  to  the  bottom  of  the  lower,  and 
one  to  that  of  the  upper  level  of  the  navigation,  as  in  the  figure 
beneath. 


The  vessel,  in  passing  from  one  level  to  the  other,  was  first 
drawn  up  on  one  of  the  inclined  planes,  and  passing  the  ridge, 
was  resisted  in  its  descent  on  the  other.  For  this  purpose  ropes 
were  passed  around  the  vessel,  and  applied  to  capstans  situated 
on  each  side. 

The  sluice,  in  its  original  form,  is  still  used  in  China,  as  is  the 
inclined  plane.  The  sluice  is  also  still  used  in  the  Low  Countries, 
and  in  the  valley  of  the  Po,  in  cases  where  the  change  of  level 
is  not  more  than  two  or  three  feet.  In  almost  all  other  instances, 
locks  have  been  substituted  for  sluices,  and  several  locks,  in  the 
cases  to  which  the  inclined  plane  was  formerly  applied. 

These  methods  of  improving  the  navigation  of  rivers,  have 
the  advantage  of  requiring  a  comparatively  small  original  cost ; 


432  OF  RIVERS.  Book  VL~\ 

they  may,  therefore,  be  used  in  countries  of  thin  population,  and 
small  wealth.  They  have  the  disadvantage  on  the  other  hand, 
that  they  are  liable  to  injury  from  floods,  while  the  navigation 
is  itself  subject  to  vicissitudes  from  variation  in  the  supply  of  the 
stream.  The  passage  of  vessels  through  sluices,  and  over  the 
original  form  of  the  inclined  plane,  is  difficult,  and  requires  the 
application  of  force  of  an  expensive  character.  For  all  these 
reasons,  it  has  been  the  uniform  result  of  experience,  that  in 
spite  of  the  great  excess  of  the  first  cost,  it  is,  wherever  the  ca- 
pital can  be  obtained,  better  to  make  a  canal  parallel  to  the  river, 
than  to  attempt  to  improve  the  navigation  in  its  own  bed.  The 
principle  on  which  the  construction  of  canals  rests,  are  contained 
in  the  following  chapter. 

419.  When  a  river  is  to  have  its  navigation  improved,  or  when 
water  is  to  be  drawn  from  it  for  supplying  pipes,  for  irrigation, 
or  any  other  useful  purpose,  it  is,  generally  speaking,  necessary 
to  ascertain  the  quantity  of  water  that  it  furnishes.  For  some 
physical  investigations,  the  whole  quantity  that  flows  in  its  bed, 
may  be  the  object  of  research.  But  for  mechanical  purposes,  it  is, 
generally  speaking,  best  to  limit  the  inquiry  to  the  minimum 
supply,  for  on  this  will  depend  the  certainty  with  which  the  stream 
can  be  depended  upon  for  most  practical  uses. 

In  a  stream  of  so  small  a  size  that  barriers  can,  without  diffi- 
culty, be  erected  across  it,  a  dam  that  interrupts  its  course,  is 
constructed  at  some  convenient  point.  In  this  a  gate  is  placed, 
formed  of  a  rectangular  frame,  and  a  shuttle  that  can  be  raised 
or  lowered  at  pleasure.  The  shuttle  being  closed,  the  passage  of 
the  water  is  interrupted,  and  it  would  rise  until  it  found  a  dis- 
charge over  the  dam  :  before  it  reaches  this  height,  the  shuttle 
is  drawn  up,  and  the  water  passes  out.  When  this  discharge  is 
just  sufficient  to  prevent  the  level  of  the  water  from  rising  far- 
ther, but  not  sufficient  to  cause  it  to  fall,  it  will  be  obvious  that 
the  gate  discharges  the  exact  quantity  of  water  that  the  stream 
furnishes. 

This  quantity  may  be  calculated  on  the  following  principles.  — 
The  area  of  the  open  part  of  the  gate  may  be  considered  as  an 
aperture  in  a  thin  plate,  and  must,  therefore,  receive  a  correction 
for  the  vena  contracta.  The  velocity,  which  will  vary  in  every 
horizontal  element  of  the  orifice,  will  have  for  its  mean,  that 
which  is  due  to  the  depth  of  the  centre  of  pressure  of  the  opening 
beneath  the  surface  of  the  fluid.  This  velocity  being  found  by 
the  principles  of  §  401,  and  the  formula  (61), 


is  to  be  multiplied  by  the  area  of  the  aperture  and  the  constant 
quantity  0.62,  §  408. 


Book 


OF  RIVERS. 


433 


In  streams  whose  section  has  an  area  of  more  than  two  square 
feet,  this  method  would  become  expensive  and  troublesome.  To 
gauge  streams  of  an  area  of  from  two  to  ten  or  twelve  square  feet, 
we  have  recourse  to  another  method. 

The  area  of  the  stream  is  carefully  measured.  Its  velocity 
may  next  be  determined  by  the  aid  of  an  apparatus,  called  from 
its  inventor,  the  Tube  of  Pitot.  A  tube  ABC  is  taken  and  bent 
until  one  of  its  branches  is  at  right  an- 
gles to  the  direction  of  the  other.  This 
tube  is  open  at  both  ends,  and  is  con- 
tracted at  the  opening  C  of  its  shorter 
branch.  It  is  placed  in  the  stream,  the 
branch  AB  in  a  vertical  position,  and 
the  horizontal  branch  is  turned  around 
until  the  stream  enters  freely  and  di- 
rectly into  the  orifice  E.  The  water 
entering  the  tube  with  a  determinate 
velocity,  has  such  a  force  as  would  cause 
it  to  rise  above  the  level  of  the  surface 
of  the  stream,  to  the  height  whence  it 
must  have  fallen  to  acquire  that  velocity. 
In  the  tube,  it  will  be  resisted  by  fric- 
tion, and  the  height  will  be  lessened. 
The  height  to  which  it  rises  may  be  de- 
termined by  applying  a  graduated  rod 
to  the  outside  of  the  tube,  if  of  glass. 
If,  however,  the  tube  be  not  transparent, 
and  it  is  usually  metallic,  a  rod,  be,  is 

placed  in  the  vertical  branch  of  such  specific  gravity  as  to  float  on 
water.  This  rod  being  graduated,  will  mark  the  rise  of  the  li- 
quid. The  corresponding  velocity  may  then  be  obtained  by  the 
usual  formula  (61.) 

' 


which  is  sufficiently  accurate,  except  when  the  velocity  is  small. 

Experiments  may  be  made  in  different  parts  of  the  section  of 
the  stream,  and  the  mean  of  the  results  taken  as  the  mean  velo- 
city. It  is,  however,  generally  speaking,  sufficient  to  measure 
the  velocity  in  this  manner,  near  the  middle  of  the  surface,  and 
thence  to  deduce  the  mean  velocity  by  the  formula  (462)  or 
(463).  . 

This  velocity,  multiplied  by  the  area,  will  give  the  discharge 
in  the  unit  of  time. 

To  use  the  tube  of  Pitot,  it  is  convenient  to  attach  it  to  a 
wooden  stand,  by  which  it  can  be  placed  in  the  bed  of  the  stream 
and  afterwards  levelled. 

55 


434 


OF  RIVERS. 


[Book  VI. 


This  instrument  is,  therefore,  limited  to  streams  of  not  more 
than  four  or  five  feet  in  depth. 

In  streams  of  greater  size,  the  velocity  of  the  surface  may  be 
measured  by  a  float.  For  this  purpose,  a  part  of  the  stream  must 
be  chosen,  where  it  flows  between  its  banks  without  eddies.  The 
area  of  its  section  may  then  be  determined  by  measuring  its 
breadth,  and  its  depth,  at  equal  intervals,  from  bank  to  bank,  as 
in  the  following  figure,  where  AB  represents  the  breadth,  and 


i> 


BE,  FG,  HI,  depths  measured  at  three  points,  dividing  the 
breadth  into  four  equal  parts.  The  area  may  then  be  considered 
as  made  up  of  the  two  triangles,  ADE,  BHI,  and  the  two  trape- 
zoids,  DFGE,  and  FHIG,  or  of  any  other  number  of  the  latter 
that  the  circumstances  may  require.  A  float  is  next  thrown  into 
the  stream,  and  the  time  in  which  it  is  carried  through  a  given 
distance,  that  is  accurately  measured  upon  Its  banks,  noted  by  a 
time-keeper.  The  velocity  in  the  unit  of  time  thence  deduced, 
is  reduced,  as  before,  to  the  mean  velocity,  and  multiplied  by 
the  area.  The  product  is  of  course  the  discharge.  The  floats 
used  in  this  operation,  may  be  constructed  in  various  modes. 
Thus  : 

A  piece  of  cork  may  have  a  weight  attached  to  it  by  a  string 
that  shall  render  their  united  mass  but  little  less  dense  than  water. 
No  more  of  the  cork  will  then  float  than  is  sufficient  to  render  it 
visible,  and  it  will  not  be  liable  to  disturbance  from  the  action  of 
the  wind.  In  a  very  short  time  it  will  acquire  the  whole  velo- 
city of  the  part  of  the  current  in  which  it  is  placed.  It  will  also, 
after  a  greater  or  less  time,  be  carried  to  that  part  of  the  stream 
that  has  the  greatest  velocity.  If  the  float,  and  the  weight  at- 
tached to  it,  have  similar  figures,  and  are  of  equal  size,  it  will  be 
obvious  that  the  velocity  assumed  by  it  will  be  the  mean  of  that 
of  the  two  horizontal  layers  of  the  stream  in  which  they  lie.  As 
this  condition  may  be  always  conveniently  adopted  in  practice, 
there  is  no  need  of  investigating  the  correction  that  would  be  due 
to  any  difference  in  these  respects.  If  the  distance  between  the 
two  bodies  be  small,  their  velocity  may,  without  sensible  error, 
be  considered  as  that  of  the  surface  of  the  stream. 

The  float  may  have  the  form  of  a  hollow  cylinder,  as  for  in- 
stance, a  piece  of  a  reed,  closed  at  one  end.  This  may  be  ballasted 


Book  Vl.~\  OF  RIVERS.  435 

by  shot  or  other  small  weights,  until  no  more  than  a  small  por- 
tion of  it  be  elevated  above  the  surface. 

If  the  length  of  the  tube  bear  a  considerable  proportion  to  the 
depth  of  the  stream,  it  will  float  in  a  position  considerably  in- 
clined to  the  vertical,  in  consequence  of  the  difference  of  the 
velocity  with  which  the  fluid  moves  at  different  depths;  and  in 
all  cases,  there  will  be  a  greater  or  less  inclination.  An  investi- 
gation of  the  nature  and  direction  of  the  forces  that  act,  may  en- 
able us  to  determine  the  velocity  at  the  surface  when  the  velocity 
of  the  reed  and  its  inclination  to  the  horizon  are  given.  Such  an 
investigation  may  be  found  in  Venturoli,  Vol.  II.  p.  218.  It  is, 
however,  better  not  to  use  this  instrument,  except  when  it  floats 
nearly  vertically,  in  which  case  the  error  is  unimportant,  and 
therefore  needs  no  correction. 

In  streams  of  still  larger  size,  a  part  is  chosen  in  which  the 
water  flows  for  a  distance  of  1  or  200  yards,  without  eddies. 
The  area  is  measured  as  above,  and  the  mean  of  several  measure- 
ments is  to  be  preferred  to  a  single  one.  The  sum  of  the  length^ 
of  the  lower  sides  of  the  triangles  and  quadrilateral  figures,  is 
taken  as  the  length  of  the  curved  section  of  the  bed.  The  area 
divided  by  this,  gives  the  quantity  R,  in  (454),  or  the  hydraulic 
mean  depth.  The  slope  of  the  surface,  or  I,  of  the  same  formula, 
is  next  obtained  by  levelling,  taking  the  difference  in  the  altitude 
of  the  surface  of  the  stream  at  two  points,  whose  distance  is  as- 
certained by  measurement.  These  two  quantities  being  given, 
the  mean  velocity  is  deduced  by  means  of  the  formulae,  (462)  or 
(463),  and  this  multiplied  by  the  area,  gives  the  discharge  in  the 
unit  of  time. 

An  instrument  used  for  a  variety  of  other  purposes,  and  called 
a  Dynamometer,  may  also  be  applied  to  measure  the  velocity  of 
a  stream.  The  essential  part  of  this  instrument  is  a  spring,  the 
quantity  of  whose  contraction,  under  pressures  of  known  inten- 
sities, has  been  determined  by  experiment.  If  a  plane  surface 
be  attached  to  such  a  spring,  by  a  cord  or  other  convenient 
method,  the  action  of  the  stream  will  compress  the  spring,  and 
the  amount  of  compression  will  measure  the  action  of  the  water, 
which  will  be  given  by  the  index  of  the  instrument,  in  some  con- 
ventional unit  of  weight.  This  action  may  also  be  considered 
as  represented  by  the  weight  of  a  prism  of  the  fluid,  whose  area  is 
equal  to  the  area  of  the  surface  acted  upon,  and  whose  altitude  is 
that  through  which  a  heavy  body  must  have  fallen  to  acquire  the 
velocity.  If  then  the  number  of  units  of  some  given  cubic  mea- 
sure of  water  that  is  equivalent  to  the  weight  to  be  calculated, 
and  divided  by  the  area  of  the  plane,  the  quotient  is  that  height ; 
from  this  the  velocity  may  be  calculated  by  the  usual  formula. 


436  or  RIVERS.  [Book  VL 

This  principle  is  not  absolutely  true,  as  will  be  seen  when  we 
treat  of  the  percussion  and  resistance  of  liquids.  It  is,  however, 
sufficiently  near  the  truth  for  all  practical  purposes. 

One  other  method  of  gauging  a  stream,  remains  to  be  men- 
tioned. It  is  applicable  to  the  case  where  it  is  traversed  by  a 
dam  whose  upper  surface  is  horizontal,  and  over  which  the  water 
discharges  itself  in  a  sheet,  forming  a  water-fall.  The  velocity 
with  which  a  fluid  passing  through  such  an  aperture  as  the  upper 
surface  of  the  dam  would  represent,  is  discharged,  is,  according 
to  the  reasonings  and  experiments  of  Du  Buat, 


in  which  expression  b  is  the  breadth  of  the  sheet"  of  water,  and 
h  its  depth,  both  expressed  in  English  feet.  The  formula  for 
the  French  metre,  is 

v=0.1895bVh\  (464) 

These  formulae  comprise  the  last  case  of  the  motion  of  liquids 
recapitulated  in  §  397. 


Book  VI]   .  OF  CANALS.  437 


CHAPTER  VII. 

Or  CANALS. 

420.  Canals  are  open  artificial  channels  formed  for  the  con- 
veyance of  water.     They  may  be  used  for  the  purposes  of  navi- 
gation ;  for  the  supply  of  those  intended  for  navigation,  in  which 
case  they  are  called   Feeders;  for  the  supply  of  cities ;  in  the 
draining  of  morasses,  and  for  irrigating  land  for  agricultural  pur- 
poses.    Canals  differ  in  character  from  rivers  ;  the  latter  have, 
by  the  long  action  of  antagonist  forces,  formed  beds,  that  gene- 
rally speaking  occupy  the  lowest  levels  in  vallies,  and  follow  the 
lines  of  greatest  slope.  The  former  have  some  conventional  slcrpe 
suited  to  their  object,  and  which  is  usually  uniform,  or  are  ab- 
solutely level  ;  they  follow  this  prescribed  line  along  the  sides  of 
hills,  or  through  their  masses,  and  are  often  carried  at  a  considera- 
ble height  over  streams,  ravines,  and  even  broad  vallies.     Rivers 
rise  from  humble  origin,  but  uniting  with  others  as  they  pro- 
ceed, and  receiving  the  discharge  from    lateral  vallies,   swell 
in  bulk  in  spite  of  the  causes  of  waste  that  affect  them  equally 
with  canals.     Canals,  on  the  other  hand,  maintain  an  uniform 
section  throughout  their  course,  and  their  volume  diminishes  in 
consequence  of  those   causes ;   while  if  collateral  supplies  are 
broyght  in,  they  need  be  no  more  than  equal  to  the  restoration 
of  what  has  been  -wasted  :  if  these  collateral  supplies  exceed  this 
amount,  means  must  be  provided  to  get  rid  of  the  excess. 

421.  Canals  take  their  rise  in  reservoirs,  either  in  the  form  of 
natural  streams  and  lakes,  or  of  artificial  basins,  that  collect  the 
surface  water.  The  fluid  contained  in  these  has  usually  a  velocity 
less  than  that  required  in  the  canal ;  and  even  if  it  be  a  rapid 
stream,  whence  the  canal  proceeds,  there  will  be  a  difference  in 
the  directions  of  the  two  currents,      Hence  the  water  in  the 
canal  will  not  at  once  assume  the  required  velocity;  and  if  the 
bed  have  a  constant  section  until  it  unite  with  the  reservoir,  the 
area  of  the  stream   will  diminish  as  it  recedes  from  the  reser- 
voir, until  the  constant  velocity  adapted  to  the  slope  and  cir- 
cumstances of  the  channel  be  attained.      This  diminution  of 
area  can  only  take  place  by  a  diminution  in  the  depth  of 'the 
water  in  the  canal.     Hence  at  the  origin  of  canals,  whose  area 
does  not  vary,  a  fall  will  be  formed,  by  which  the  area  of  the 
stream  will  be  diminished.     If  then  it  be  important  that  the 


438  OF  CANALS.  [Book  VI. 

canal  shall  be  filled,  it  must  spread  out  as  it  approaches  the  reser- 
voir. If  we  suppose  it  to  be  influenced  by  the  same  causes  that 
affect  the  vena  contracta,  and  that  the  channel  has  a  rectangular 
section  of  uniform  depth,  the  plan  of  the  channel  would  be  form- 
ed on  each  side  by  a  logarithmic  curve,  whose  axis  and  greater 
and  less  ordinates  have  the  proportion  of  5  :  6.25  :  4  ;  the  second 
number  being  half  the  breadth  of  the  channel  at  the  reservoir ; 
and  the  third  half  its  uniform  breadth.  If  the  figure  were  investi- 
gated on  the  principle  that  the  velocity  of  water  moving  in  a 
channel  varies  inversely  as  its  area,  it  would  be  found  that  the 
proper  figure  of  the  banks  would  be  a  parabola,  whose  vertex  is 
at  the  point  where  the  breadth  of  the  channel  becomes  constant. 
Neither  of  these  methods  of  investigation  is  free  from  objection  ; 
but  it  is  evident  from  experiment  and  observation,  that  in  order 
that  water  shajl  enter  into  a  channel  without  forming  a  fall,  or 
that  it  shall  completely  fill  it,  the  channel  must,  at  the  reservoir, 
have  a  width  greater  than  the  breadth  of  the  uniform  section  that 
it  has  at  other  points  ;  and  this  increase  of  breadth  should  take 
place  in  the  form  of  a  curve  convex  towards  the  axis  of  the 
canal. 

Such  a  form  is  to  be  found  in  nature  when  streams  take  their 
rise  in  lakes,  or  other  similar  reservoirs.  If  a  canal  be  formed 
in  soft  earth,  it  will  gradually  wear  away  the  earth  until  it  as- 
sume such  a  form  ;  but  in  solid  rock  such  a  shape  cannot  be  spon- 
taneously assumed.  Even  in  canals  made  in  soft  earth,  it  is  bet- 
ter to  give  the  requisite  shape  artificially,  than  to  wait  for  the 
slow  action  of  the  water. 

422.  The  shape  of  the  section  of  a  canal  is  usually  a  trapezium, 
two  of  whose  sides  are  parallel  and  horizontal ;  the  other  two 
equally  inclined  to  the  horizon.  This  inclination  will  depend 
upon  the  nature  of  the  soil  in  which  the  canal  is  formed,  being 
least  in  tenacious  earth,  and  greatest  in  loose  soils.  No  soil  will 
maintain  itself  when  the  base  of  the  slope  is  less  than  one  and  a 
third  times  its  height,  or  in  the  proportion  of  four  to  three.  While 
in  loose  soils  the  slope  must  be  at  least  as  great  as  in  the  propor- 
tion of  two  to  dne  ;  or  the  base  twice  as  great  as  the  height.  The 
force  that  acts  upon  the  bank  is  the  pressure  of  the  water,  and 
this  is  partly  exerted  to  push  the  bank  aside, and  partly  to  over- 
turn it  by  a  rotary  motion.  In  banks  of  earth,  where  the  mate- 
rial has  little  cohesive  strength,  the  former  effort  is  most  likely 
to  be  injurious ;  while  in  masses  of  masonry,  the  latter  is  the  more 
important 

To  investigate  the  proper  thickness  of  a  bank,  whose  height 
and  slope  and  the  material  of  which  it  is  composed  are  given  : 


Book  ft.]  OF  CANALS.  439 

Let  ABCD  be  the  section  of  a  prismatic  mound  of  earth, 
pressed  by  water  on  the  side  AB  ;  let  the  slopes  on  each  side  be 

A  B  equal,  and  suppose  the  bank 

itself  to  be  composed  of  mate- 
terials  of  infinite  cohesive  force, 
but  capable  of  being  moved 
horizontally  in  the  direction 
AD,  by  a  force  equal  to  the 
friction  among  its  particles. 
B  B  e.  c 

Let  AB,  the  length  of  the  face  =  a ; 

BAE,  the  base  of  the  slope,  =  b ; 

ABE,  the  vertical  height,  =  h ; 

the  angle  of  the  slope,  ABC,  =  i  ; 

AD,  the  thickness  of  the  top  of  the  bank  =  x  ; 

the  density  of  the  material,  that  of  water  being  unity,    —  D  ; 
the  co-efficient  of  the  friction  =  /; 

the  thickness  of  the  bank  at  bottom  will  be  =  x  +  2  b  ; 

The  pressure  on  the  line  AB  will  be,  §  331, 

la/t; 

this  force  acts  at  the  centre  of  pressure  in  a  direction  per- 
pendicular to  the  face  of  the  bank.  If  then  it  be  resolved  into 
two  components,  one  in  a  horizontal,  the  other  in  the  vertical 
direction,  the  former  only  will  tend  to  thrust  the  bank  from  its 
plane  ;  the  latter  will  add  to  the  weight  that  presses  on  the  base, 
will  increase  the  friction,  and  consequently  add  to  the  stability. 
The  first  of  these  components  will  be,  §  13, 

^  ah  sin.  i ; 
the  second, 

\  ah  cos.  i ; 
but  as 

h  b 

sin.  i—~,  and  cos*  i——  , 
••4   .  ct>  a 

these  forces  become  respectively 

i/i3,  and  %hb\ 

of  which  that  represented  by  £  /i2,  tends  to  thrust  the  bank  from 
its  place. 

This  force  is  resisted  by  the  friction  on  the  base,  which  is  a 
function  of  the  whole  pressure,  and  this  is  made  up  of  the  weight 
of  the  bank,  and  the  force  \  h  b  derived  from  the  liquid  pressure 
acting  downwards. 

The  weight  of  the  bank  is  found  by  multiplying  its  area  by  its 
density,  and  is 


OP  CANALS.  [Book 

we  therefore  have,  as  the  condition  of  equilibrium, 


(465) 
whence  we  obtain  for  the  value  of  2-, 


-• 

slope 

(467) 


When  the  bank  is  triangular,  #=0,  and  the  base  of  the  slope 
becomes 


to  find  the  co-efficient/,  we  have 

<468> 


The  mean  density  of  earthy  substances  being  about  2,  we  have 
in  the  case  where  26=3/i, 

f=T2_=  0.133;    • 
and  when  6 =2 A, 

/=A; 

and,  as  the  pressure  at  the  surface  is  evanescent,  and  a  triangular 
mound  would  of  course  resist  it,  these  values  may  be  applied  in 
other  investigations. 

In  practice,  however,  the  water  will  enter  into  the  pores  of 
the  earth,  and  thus  the  surface  exposed  to  pressure  will  be  much 
increased. 

To  allow  for  this,  let  the  co-efficients /be  reduced  to  one  half, 
or  in  firm  earth  to  0.06,  and  in  loose  earth,  to  0.05.  We  shall 
then  have,  in  the  latter  case,  in  a  bank  of  triangular  section, 

and  the  whole  base  of  the  bank, 

26=8/i.  (469) 

When  the  mass  of  the  bank  is  given,  as  that  part  of  the  liquid 
pressure  which  tends  to  thrust  the  bank  aside,  diminishes  as  the 
inclination  of  the  face  on  which  it  acts  increases ;  while  on  the 
opposite  side,  loose  earth  supports  itself  at  a  slope  whose  base  is 
to  the  height  as  2  :  1 :  we  may  infer,  that  the  maximum  of  strength 
will  be  attained  when  the  slope  on  which  the  water  presses  has 
for  its  base 

6=6 /i; 

and  when  the  base,  6',  of  the  opposite  side,  is 

In  loose  earth,  when  the  bank  has  a  trapezium  for  its  section, 
and  6=2/?,  we  obtain  for  the  value  of  #,  from  (466), 

(470) 


Book  J7.]  OP  CANALS.  441 

in  firm  earth,  where  6=1  i  h, 

x=2,3(K).  (471) 

These  give  the  thickness  of  the  bank  at  the  surface  of  the  water  ; 
but  as  it  is  usual  to  make  the  bank  of  a  canal  one  foot  higher 
than  the  level  of  the  water  it  contains,  we  would  have  for  the 
thickness  at  its  upper  surface,  in  the  case  of  loose  earth,  in  feet, 

x=2.5(h) — 4,  (472) 

and  in  firm  earth, 

o?=2.3(/i)— 3.  (473) 

423.  A  canal  is  usually  confined  between  a  bank  on  one  side, 
whose  dimensions  may  be  determined  on  the  foregoing  princi- 
ples, and  a  towing  path,  the  breadth  of  whose  upper  surface  must 
be  sufficient  for  a  road,  on  which  the  animals  employed  in  draugfit 
may  easily  pass  each  other.  If,  then,  the  dimensions  deduced 
above  be  not  sufficient  for  this  last  object,  the  breadth  of  the  up- 
per surface  of  the  towing  path  must  be  increased  to  at  least  nine 
feet.  For  the  other  bank,  the  .usual  rule  is,  to  make  its  breadth 
at  top  equal  to  the  height  measured  from  the  bottom  of  the  canal ; 
but  in  this  case  there  should  be  abermoffrom  1  to  1£  feet  at  the  le- 
vel of  the  water,  by  which  the  thickness  of  the  bank  at  the.  water's 
edge  will,  in  usual  cases,  nearly  coincide  with  our  formulae,  (470) 
or  (471,  and  which  will  have  the  advantage  .of  preventing  the 
wash  of  the  banks  from  falling  into  thelcanal. 

To  prevent  the  entrance  of  rain  water,  a  ditch  called  the  coun- 
ter ditch  is  formed  on  the  outside  of  each  of  the  banks.  This  is 
particularly  necessary  in  side  lying-ground,  where  the  rain  may 
produce  injurious  effects. 

The  profile  of  a  well-constructed  canal  will  therefore  present 
the  following  figure : 


And  when  the  breadth  and  depth  are  given,  the  relations  between 
the  depth  of  excavation  and  the  height  of  embankment,  that  will 
just  suffice  to  give  the  proper  form,  is  a  simple  geometric  problem. 

424.  Canals  are  applied  to  the  purposes  of  inland  navigation 
in  several  cases  : 

(1.)  As  has  been  already  stated,  they  may  be  employed  to  pass 
obstacles  upon  rivers,  that  are  in  other  respects  navigable,  or  may 
be  constructed  parallel  to  a  stream  whence  they  derive  their  sup 
ply  of  water. 

56 


442  OF  CANALS.  [Book  VI. 

(2).  They  may  be  made  to  communicate  between  two  naviga- 
tions of  eqaal  levels,  drawing  their  supply  from  both,  or  between 
two  of  different  heights,  drawing  their  supply  from  the  higher. 

(3).  They  may  form  a  communication  between  two  naviga- 
tions, passing  over  ground  higher  than  either.  This  is  at  present 
the  most  usual  case  of  canal  navigation.  Such  canals  are  said  to 
have  a  summit  level,  or  to  be  a  point  de  partage. 

425.  The  dimensions  of  navigable  canals  will  depend  upon  the 
section  of  the  vessels  intended  to  navigate  them.     They  must  in 
the  first  place  be  wide  enough  to  permit  two  vessels  to  pass  each 
other  with  freedom  ;  for  this  purpose  the  breadth  at  bottom  is 
usually  made  twice  as  great  as  the  breadth  of  beam  of  the  vessels  j 
in  the  second  place,  the  depth  is  usually  made  one  foot  more  than 
their  draught  of*water.     This  is  done  for  two  reasons  :  first,  be- 
cause it  is  found  that  vessels  are  much  more  impeded,  when  the 
channel  in  which  they  move  is  shallow,  than  when  it  is  deeper  ; 
and  secondly,  in  order  to  pro  vide  for  any  accidental  defect  of  wa- 
ter, or  for  deposits  of  earth  thatmay  lodgein  the  canal.  In  eitherof 
these  cases,  the  vessels  would  be  exposed  to  take  the  ground, 
were  there  not  an  extra  depth  of  water. 

426.  The  bed  of  a  canal  must  be  absolutely  level,  or  have  no 
more  slope  than  is  sufficient  to  convey    water  to  replace  that 
which  may  be  wasted.    Hence,  when  the  navigations  it  is  to  con- 
nect are  of  different  levels ;  when  constructed  on  the  bank  of  a 
stream  whose  surface  must  of  course  slope  ;  or  when  it  surmounts 
a  summit,  it  must  be  made  in  a  series  of  channels  at  different 
levels,  and  means  must  be  provided  to  pass  vessels  from  one  level 
to  another.     That  which  is  most  usually  employed  is  called  a 
Lock. 

A  lock  is  a  four-sided  chamber,  contained  between  two  parallel 
walls  in  the  direction  of  its  length,  and  by  two  gates  or  sluices, 
situated  at  the  two  extremities. 

The  bottom  of  a  lock  is  a  floor  of  wood,  or  masonry,  on  the 
same  level  with  the  bottom  of  the  lower  of  the  two  reaches  of 
the  canal  that  it  serves  to  connect.  The  gates  rise  to  the  height 
of  the  water  in  the  upper  reach,  and  the  walls  to  the  height  of  the 
banks  of  that  reach.  The  lower  gate  reaches  to  the  floor  of  the 
lock,  the  upper  gate  is  usually  established  upon  a  transverse  wall, 
the  top  of  which  is  on  a  level  with  the  bottom  of  the  upper  reach, 
and  which  is  called  the  Breast  Wall. 

When  the  water  in  the  lock  is  upon  a  level  with  the  surface 
of  the  water  in  the  higher  reach,  it  is  said  to  be  full ;  when  on 
a  level  with  that  in  the  lower  reach  of  the  canal,  it  is  said  to  be 
empty.  The  gates  are  suspended  from  the  walls  in  such  a  man- 


Book  F7.] 


CANALS. 


443 


ner.  as  to  close,  when  they  undergo  a  pressure  from  above  ;  they 
may  be  opened  when  the  pressure  on  each  side  is  equal. 
A  plan  and  section  of  a  lock  are  represented  beneath. 
FIG.  2.  FIG.  1. 


CO 


Fig.  1st,  is  a  longitudinal  section  ;  fig.  2d,  a  horizontal  plan 
of  a  lock. 

A,  B,  C,  D,  are  the  walls  of  which  E  and  F  are  the  parts  that 
enclose  the  chamber. 

M,  G,  N,  D,  the  breast  wall  on  which  the  upper  gate  is  sup- 
ported. 


444  OF  CANALS.  [Book  VI 

HH,  recesses  in  which  the  upper  gate  is  received  when  open. 

1 1,  similar  recesses  to  receive  the  lower  gate. 

K,  sill  of  the  upper  gate. 

L,  sill  of  the  lower  gate. 

O,  upper  opening  of  the  culverts  that  form  a  communication 
between  the  waters  of  the  upper  and  lower  levels  of  the  canal,  in 
order  to  fill  the  lock. 

V,  vault  in  the  breast  wall  into  which  these  culverts  enter. 

WW,  level  of  the  water  in  the  upper  level  of  the  canal. 

WW,     do         "  ^     lower     do. 

The  operations  of  filling  the  lock  from  the  upper  level  of  the 
canal,  and  of  emtying  it,  by  a  discharge  into  the  lower  level, 
may  be  effected  by  means  of  small  gates,  or  wickets,  sliding  in 
grooves  in  the  timbers  of  the  gates. 

\T<*  this  method  no  objections  apply  in  the  lower  gate  ;  but  in 
the  apper,  the  water  spouting  from  the  top  of  the  breast  wall  may 
injure  the  lock,  or  enter  into  the  vessel  contained  in  it,  which 
may  thus  be  destroyed,  or  sunk.  Hence,  passages  called  Culverts, 
have  been  formed  in  the  masonry  of  the  walls  of  the  lock,  to 
which  wickets  are  adapted  ;  these  passes  opening  from  the  water  in 
the  upper  level,  are  inclined  iri  such  a  manner  as  to  discharge 
themselves  below  the  surface  of  the  water  in  the  lock,  even  when 
as  much  as  possible  is  drawn  off.  In  the  Canal  du  Centre,  in 
France,  the  culverts  descend  vertically,  and  discharge  themselves 
into  an  open  vault,  formed  by  the  thickness  of  the  breast  wall. 
A  lock  has  recently  been  invented  in  this  country  by  Mr.  Can- 
vass White,  in  which  the  breast  wall  is  omitted,  and  the  upper 
gate  is  as  tall  as  the  lower  one.  In  this  form,  the  culverts  in  the 
walls  become  unnecessary,  and  the  breast  wall  which  is  for  many 
reasons,  the  weakest  part  of  a  lock,  is  suppressed.  The  bottom  of 
the  upper  and  lower  levels,  are  united  by  a  slope,  rising  from 
the  sill  of  the  upper  gate. 

427.  When  a  vessel  is  to  rise  from  the  lower  to  the  upper  level  of 
the  canal,  it  may  either  find  the  lock  full,  or  empty  ;  in  the  for- 
mer case,  the  surplus  water  must  be  discharged  through  the 
wickets,  until  it  reach  a  common  level  on  each  side  of  the  lower 
gate  ;  in  the  latter,  the  water  in  the  lock,  and  in  the  lower  level 
are  already  at  an  equal  height.  The  pressure  being  therefore 
equal  on  each  side  of  the  lower  gate,  it  may  be  opened,  and  the 
boat  drawn  forward  into  the  lock.  The  lower  gates  are  next  shut 
behind  it,  and  the  wickets  or  culverts  that  communicate  with  the 
water  above  opened.  This  will  therefore  pass  into  the  lock,  and 
as  it  finds  no  exit  through  the  lower  gate,  will  fill  the  lock  ;  as  the 
lock  fills,  the  vessel,  buoyant  on  the  surface,  rises;  the  filling  of 
the  lock,  and  rise  of  the  vessel  continue,  until  the  water  stand  on 


Book  F/.]  op  CANALS.  445 

each  side  of  the  upper  gate,  at  a  common  level.  The  pressure  on 
each  side  of  the  upper  gate  being  then  equal,  it  may  be  opened, 
and  the  vessel  drawn  forward  into  the  upper  level  of  the  canal. 

If  the  canal  be  empty  when  a  boat  is  to  descend,  it  must  be  fill- 
ed, or  if  full,  kept  so  until  the  upper  gate  be  opened,  and  the  ves- 
sel admitted.  The  upper  gate  is  then  closed  behind  the  vessel, 
and  the  water  discharged  from  the  lock,  through  the  wickets  of 
the  lower  gate,  until  the  water  within,  and  in  the  level  below, 
reach  the  same  height,  when  the  lower  gate  may  be  opened,  and 
the  vessel  drawn  out. 

The  lock  appears  to  have  taken  its  origin  from  the  accidental 
juxta-position  of  two  sluices,  in  the  action  of  which  its  important 
and  valuable  properties  were  discovered.  It  seems  to  have  been 
first  intentionally  used  in  the  Canal  of  Martizana,  in  Italy,  about 
the  eleventh  century.  '**;; 

428.  To  determine  the  thickness  of  the  longitudinal  walls  that 
confine  a  lock,  when  the  depth  of  water,  and  the  nature  of  the 
material  is  given,  we  must  in  the  first  place  consider,  that  being 
built  of  masonry,  the  resistance  to  lateral  thrust,  is  that  of  the 
friction  of  stone  upon  stone,  at  the  joints,  and  of  the  cohesive  force 
of  the  stone  at  other  points  ;  the  former  is  aided  by  the  cohesive 
force  of  the  mortar,  and  these  resistances  being  both  great,  the 
water  will  exert  a  more  powerful  influence  to  overturn  the  wall, 
than  to  move  it  laterally.  As  the  pressure  at  the  surface  of  the 
water  is  0,  and  as  the  wall  may  be  built  in  a  vertical  position,  we 
may  assume  it,  for  the  purpose  of  investigation,  to  have  a  section 
of  the  figur^  of  a  right  angled  triangle. 

Let  ABC,  be  a  section  of  the  wall ;  let  the  height  AB,    =p ; 
the  thickness  BC,  =#; 

v*  the  liquid  pressure  on  the  face  AB,  will 

be,  §417,    ' 

and  it  will  act  to  overturn  the  wall  at  the 
point  P,  the  centre  of  pressure,  with  a 
moment  of  rotation  represented  by 


The  resisting  force  will  be  the  weight 

\  **  •* 

;B  "~  C       and  to  find  its  moment  of  rotation  it 

must  be  multiplied  by  the  perpendicu- 
lar distance  of  its  direction  from  the  point  C,  which  is  |  x.  This 
moment  of  rotation,  then,  will  be 


and  in  the  case  of  equilibrium  between  the  two  forces, 


446  or  CANALS.  [Book  VI. 

\#=\V*\  (474) 

whence  we  obtain 


if  we  take  the  density  of  the  materials  to  be  2f  we  have 

x=±p  ,  (476) 

or  the  thickness  of  the  wall  at  bottom,  should  be  equal  to  half  its 
height. 

In  the  construction  of  locks,  the  thickness  of  the  walls  at  bottom  is 
made  equal  to  half  the  depth  of  the  water  they  contain  when  full. 
But  the  top  of  the  wall  is  higher  than  the  top  of  the  gates,  which 
determines  the  highest  level  of  the  water,  in  order  to  prevent  any 
accidental  overflow  ;  and  the  section  of  the  wall  is  not  triangular, 
but  quadrilateral.  The  top  must  have  a  sufficient  thickness  to 
admit  of  the  masonry  being  carried  up  in  two  faces  of  ashler, 
which  are  filled  up  within  by  irregular  pieces,  laid  in  water  proof 
cement,  and  grouted,  in  order  to  prevent  any  filtration.  This 
thickness  cannot  be  less  than  from  3  to  4  feet.  When  the  wall  is 
of  brick,  it  may  be  of  less  thickness  at  top,  but  must  be  covered 
with  a  coping  strongly  clamped.  When  the  depth  of  water  to  be 
supported  is  less  than  7  or  8  feet,  the  wall  will  have  two  parallel 
faces  ;  at  greater  depths  the  outer  face  slopes,  until  it  has  a  thick- 
ness at  bottom  of  half  the  depth  of  water. 

The  walls  of  the  lock  are  continued  at  its  two  ends,  until  they 
meet  the  banks  of  the  canal  ;  and  as  the  latter  is  wider  than  the  lock, 
these  portions  diverge,  and  are  called'  the  Wing  Walls;  those  at 
the  lower  end  of  the  canal  decrease  in  height,  until  they  meet  the 
earthen  bank.  These  walls  are  represented  in  the  profile,  and 
section  at  ADandBC. 

.  These  walls  having  less  stress  to  undergo  than  those  of  the 
chamber  of  the  lock,  need  not  be  thicker  at  bottom  than  one  third 
of  their  own  height. 

429.  The  gates  of  locks  are  frames  of  timber,  covered  with 
plank  ;  the  lower  gate  reaches  from  4he  bottom  of  the  lock  ;  the 
upper  from  the  top  of  the  breast  wall,  to  the  level  of  the  water  in 
the  upper  reach  of  the  canal. 

As  frames  of  a  quadrilateral  figure  are  liable  to  change  their 
shape  when  suspended  at  one  end,  the  frame  must  have  diagonal 
braces,  or  the  plank  may  be  put  on  in  a  diagonal  direction.  The 
gate  is  suspended,  by  adapting  a  gudgeon  to  the  bottom  of  one  of 
its  uprights,  and  passing  an  iron  collar  built  into  the  wall  over  its 
upper  end.  This  post  must  therefore  project  below  the  bottom 
of  the  gate,  and  a  circular  cavity  is  made  in  the  bottom  of  the 
lock,  to  receive  its  gudgeon.  This  post,  as  well  as  that  on  the 


Book  VL~\ 


OF  CANALS. 

, 


447 


opposite  side  of  the  gate,  is  sufficiently  tall  to  rise  above  the  top 
of  the  wall,  in  order  to  be  framed  into  the  lever  that  serves  to 
open  and  shut  the  gate.  This  lever  extends,  when  the  gate  is 
shut,  over  the  top  of  the  wall,  and  serves  as  a  counterpoise  to  the 
gate.  The  form  of  the  gates  will  be  understood  by  reference  to 
the  figure. 


// 

.f; 

'AWJ\VA 

\YAWA\\\\, 

'A\\YA\\\V/ 

\YA\WA\l 

mrmr/ 

The  bottom  of  the  lock  is  lower,  within  the  space  where  the 
gates  swing,  than  at  any  other  place,  and  the  gates  when  shut, 
rest  against  a  sill.  The  walls  also,  are  built  in  such  a  form  as  to 
admit  the  gate  into  their  thickness,  when  open,  in  such  manner 
that  its  face,  and  that  of  the  wall,  shall  then  be  in  the  same  plane. 
The  outer  side  of  the  post  on  which  the  gate  hangs  is  rounded,  and 
the  stones  are  cut,  where  it  touches  the  wall,  into  a  curved  hollow, 
that  permits  its  motion,  and  to  which  the  retired  part  of  the  wall 
is  a  tangent. 

When  the  lock  has  a  small  breadth,  say  not  exceeding  four 
feet,  both  of  the  gates  may  be  formed  of  a  single  leaf,  and  are, 
when  shut,  at  Bright  angles  to  the  faces  of  the  walls.  When  the 
breadth  does  not  exceed  six  or  seven  feet,  the  upper  gate  may 
still  be  in  a  single  leaf;  but  in  this  case  the  lower  gate,  and  in 
all  locks  whose  breadth  exceeds  seven  feet,  both  gates  must  be 
formed  of  two  equal,  and  similar  leaves,  suspended  from  the  op- 
posite sides  of  the  lock.  The  reason  of  this  is,  that  the  pressure 
would  then  become  too  great  to  be  borne  by  a  single  length  of 
timber,  supported  at  one  end. 

430.  When  lock  gates  are  made  of  two  leaves,  they  must  be 
so  placed  as  to  afford  each  other  mutual  support.  For  this  pur- 
pose, instead  of  shutting  in  such  a  manner  as  to  lie  in  one  plane, 


443  OF  CANALS.  [Book  VI. 

the  two  leaves  make  an  angle  with  each  other,  and  the  sill  has  the 
figure  of  an  isosceles  triangle,  whose  vertex  is  turned  towards  the 
upper  level  of  the  canal.  This  arrangement  will  be  seen  in  the 
draught,  on  page  443. 

It  will  be  obvious,  that  if  this  angle  be  a  right  angle,  the  tim- 
bers of  each  leaf  will  receive  the  pressure  of  the  other  in  the  di- 
rection of  their  length,  and  will  therefore  afford  the  greatest 
mutual  support ;  while  if  they  lie  in  one  plane,  they  will  not  give 
each  other  any  support.  But  in  the  former  case,  the  timbers  will 
be  longer,  and  have  less  strength  to  bear  the  liquid  pressure  that 
acts  upon  the  leaf  of  which  they  are  a  part,  than  in  the  latter. 
Upon  these  principles,  the  proper  angle  at  which  the  gates  should 
meet,  in  order  to  possess  a  maximum  of  strength,  may  be  investi- 
gated. 

The  pressure  of  water  upon  the  leaf  AC,  when  the  depth  is 
constant,  will  be  proportioned  to  the  length  of  AC,  which  we  shall 


call  / ;  and  the  strength  of  the  timbers  which  form  the  gate  will 
be  inversely  proportioned  to  I ;  call  the  pressure  on  the  unit  of 
surface,  P ;  and  S,  the  respective  strength  of  the  material.  We 
have  in  the  case  of  equilibrium, 


and 

S=P/2. 

The  strength,  then,  in  order  to  resist  the  pressure,  must  be  a 
constant  function  of  the  square  of  the  horizontal  dimension  of  the 
leaf;  or  if  we  call  the  breadth  of  the  lock  2a,  and  the  projection 
CD  of  the  sill  #,  a  constant  function  of  (a2+^)« 

The  leaf  AC,  also  derives  strength  from  the  other  leaf  BC  :  if 
we  decompose  this  force  into  two  parts,  one  of  which  is  perpen- 
dicular, the  other  parallel  to  AC,  and  which  may  be  represented 
by  CE,  EB  ;  that  represented  by  EB,  will  alone  act  to  support 
the  leaf  AE.  But  this  force  will  vary  with  the  sine  of  the  angle 
A,  while  x  varies  with  the  tangent ;  it  will  therefore  be  a  constant 


V  \ 

Book  rf.]  OF  CANALS.  449 

'  .     * 

function  of  ax.  The  whole  resistance  which  a  gate  of  a  given 
breadth  will  oppose  to  the  pressure  of  the  water,  will  decrease 
with  the  former  of  these  quantities,  and  increase  with  the  latter, 
of  both  of  which  it  is  a  constant  friction,  or  will  be  greatest  when 
ax — (a2+#2)  is  a  maximun,  or  when 

x=-a=asia.  30°. 

Hence,  the  greatest  strength  will  be  attained  when  the  pro- 
jection CD  of  the  sill  is  one  fourth  of  the  breadth  of  the  lock, 
and  the  angle  that  the  gate  makes  with  the  wall  of  the  lock  is 
60°. 

When  the  length  of  the  horizontal  timbers  that  form  the  leaves, 
of  a  gate,   is  determined  by  means  of  the  breadth,  and  the  most 
advantageous  projection  thus  investigated,  their  dimensions  may 
be  calculated  by  considering  them  as  beams  supported  at  each 
end,  upon  the  principles  of  §  189. 

431.  The  height  which  is  given  to  locks,  is  the  sum  of  two 
quantities  :  the  depth  of  the  water  in  the  lower  level  of  the  canal, 
and  the  difference  between  the  level  of  the  two  surfaces.  The 
latter  is  called  the  Fall  of  the  Lock.  The  fall  that  is  most  advan- 
tageous, will  depend  upon  a  variety  of  circumstances. 

The  nature  of  the  ground,  in  some  cases,  prescribes  ,the 
proper  fall ;  but  in  the  hands  of  a  skilful  engineer,  the  quantity 
of  fall  may,  generally  speaking,  be  settled  upon  other  principles. 

The  fall  of  the  locks  of  a  given  canal  should  be  constant,  so 
that  the  water  discharged  from  one  may  just  suffice  to  sup- 
ply those  below  it.  In  this  case,  the  successive  levels  should 
be  each  capable  of  containing  one  lock  full  of  water,  in  addi- 
tion to  that  which  is  necessary  for  the  navigation,  without  over- 
flow. The  service  of  the  canal  would  then  be  performed  without 
interruption,  and  without  waste.  This  rule,  however,  neglects 
the  loss  of  water  by  evaporation  arid  leakage :  a  better  principle 
would  be,  that  the  fall  of  the  several  locks  should  decrease  from 
the  point  at  which  the  canal  receives  water,  until  a  new  supply 
be  admitted,  when  they  are  again  to  be  restored  to  their  original 
height. 

It  is  next  to  be  remarked  that  locks  of  great  fall  expend  a  great 
quantity  of  water,  and  can  therefore  be  used  only  when  that  is 
abundant. 

In  locks  of  small  fall,  boats  require  nearly  as  much  time  to  pass 
each  of  them,  as  to  pass  one  of  greater  height.  Hence,  when  a 
given  fall  is  distributed  among  several  locks,  instead  of  a  single 
one,  the  delays  are  much  increased. 

The  thickness  of  the  walls  increasing  with  the  depth  of  water, 

57 


450  or  CANALS.  [Book  VI. 

with  which  their  height  increases  also,  the  cost  of  masonry  wll 
increase  with  the  square  of  the  depth.  This  depth  is  not  the 
fall  of  the  lock,  but  is  the  sum  of  that  quantity,  and  the  constant 
depth  of  the  canal.  Thus  the  whole  cost  of  the  masonry  of  a  set 
of  locks,  to  overcome  a  given  fall,  will  decrease,  with  the  increase 
of  their  fall,  and  diminution  of  their  number,  to  a  certain  limit, 
easily  determined  in  each  particular  case,  and  afterwards  increase. 
In  this  investigation,  other  circumstances  will  come  into  view, 
such  as  the  cost  of  wood  and  iron  work  ;  the  expense  of  making 
and  securing  the  foundations. 

Locks  are  liable  to  danger' from  the  filtration  of  water  through 
the  earth  in  which  they  are  placed:  this  may,  in  some  cases,  be 
so  great,  that  the  bottom  of  the  lock  may  be  pressed  upwards  by 
a  liquid  pressure,  due  to  the  whole  surface  of  the  lock  and  the 
head  of  water  in  the  upper  level.  •  To  counteract  this  pressure,  we 
have  the  weight  of  the  lock  itself,  and  that  of  the  water  it  contains. 
In  a  high  lock,  when  empty,  the  former  force  may  preponderate, 
and  the  lock  be  forced  upwards. 

Taking  all  these  circumstances  into  account,  it  has  been  laid 
down  as  a  rule,  that  the  fall  of  locks  should  not  exceed  ten,  nor 
fall  short  of  eight  feet. 

432.  The  length  of  a  lock  will  depend  upon  that  of  the  largest 
vessels  that  are  to  pass  it,  and  will  be  such  that  a  vessel  can  lie, 
without  risk  of  touching,  between  the  angle  of  the  Isaves  of  the 
lower  gate  and  the  breast  wall. 

The  breadth  of  locks  must,  in  like  manner,  be  adapted  to  that 
of  the  vessels,  and  is  usually  made  one  foot  greater  than  their 
breadth  of  beam.  So  also,  as  has  been  seen,  the  dimensions  of 
the  canal  itself  will  depend  upon  that  of  the  vessels  that  are  to 
navigate  it. 

In  determining  the  appropriate  size  of  vessels,  the  nature  of  the 
adjacent  navigable  communications  must  be  taken  into  account, 
as  well  as  the  expense  of  construction,  and  the  character  of  the 
moving  power  employed  upon  them.  As  a  general  rule,  the  ex- 
pense of  construction  increases  even  in  a  higher  ratio  than  the 
dimensions  of  the  canal ;  hence,  when  it  is  intended  to  connect 
two  navigations  already  in  existence,  it  should  be  calculate^  for 
the  smallest  vessels  that  usually  ply  upon  them. 

The  moving  power  employed,  is  usually  that  of  horses,  walk- 
ing upon  the  towing  path  of  the  canal ;  and  a  loss  of  power  would 
ensue  if  the  boats  were  smaller  than  is  calculated  for  the  draught 
of  a  single  horse.  It  has  been  found,  by  long  experience,  that  a 
horse  readily  drags  twenty-five  tons,  in  a  boat  weighing  five  tons, 
upon  a  canal,  or  thirty  tons  in  all.  Boats  of  this  tonnage  are, 
therefore,  well-adapted  to  canal  navigation,  and  may  have  a 


Book  VI.']  OF  CANALS. 

length  of  about  60  feet,  a  breadth  of  beam  of  8  feet,  and  a  draught 
of  water  of  4  feet.  On  the  other  hand,  it  is  to  be  considered, 
that  the  resistance  to  boats  of  similar  figure,  increases  with  the 
area  of  their  midship  frame,  or  in  a  ratio  no  higher  than  the  square 
of  their  homologous  dimensions ;  while  the  tonnage  they  carry 
increases  with  the  cube :  and  that  large  vessels,  on  a  canal,  do 
not  require  a  greater  number  of  hands  to  navigate  them  than 
small  ones.  Two  are,  in  all  cases,  sufficient ;  the  one  to  steer 
the  boat,  the  other  to  drive  the  horse. 

433.  In  the  earlier  canals,  it  was  a  frequent  custom  to  build 
locks  in  juxta-position,  like  steps  of  stairs,  the  upper  gate  of  the 
one  answering  as  the  lower  gate  of  the  next,  and  so  on.  This 
disposition  is  faulty.  (1.)  Inasmuch  as  it  causes  a  greater  use  of 
water.  And  (2.)  Because  it  delays  the  navigation. 

When  the  locks  have  a  space  between  them,  sufficient  for  two 
boats  to  pass  each  other,  and  all  the  locks  are  of  the  same  height, 
a  single  lock  full  of  water  will  suffice  for  the  passage  of  a  boat 
from  the  summit  to  the  lowest  level'of  the  canal ;  and  if  the  trade 
exactly  alternate,  the  ascending  boats  will  find  the  locks  empty; 
and  they  will  be  filled,  in  readiness  for  the  descending  boats,  by 
the  operation  of  raising  those  that  are  ascending.  When  the  locks 
are  in  juxta-position,  the  first  descending  boat  passes  through  the 
system  with  but  one  lock  full  of  water,  but  when  another  is  to  rise, 
it  finds  all  the  locks  empty ;  and  hence,  there  must  be  drawn  from 
the  higher  level  as  many  locks  full  of  water  as  there  are  locks  in 
the  system.  The  second  descending  boat  finds  all  the  locks  full, 
and  the  whole  of  them  must  be  discharged  in  order  to  permit  it 
to  pass.  It  is  therefore  evident,  that  when  the  locks  are  combined 
in  a  system,  they  use  a  quantity  of  water  as  many  times  greater 
than  when  they  are  separate,  as  there  are  locks  in  the  system. 
This  waste  will  not  be  obviated  wholly,  by  merely  placing  a  basin 
between  the  locks,  in  which  two  boats  may  pass  each  other  ;  but 
this  basin  must  be  large  enough  to  receive  and  retain  a  lock  full 
of  water,  and  to  permit  that  same  quantity  to  be  drawn  from  it 
without  rendering  the  water  too  shallow.  If  the  basin  have  not 
sufficient  capacity,  a  part  of  the  descending  water  will  run  to 
waste,  and  the  boats  may  take  the  ground.  If  the  circumstances 
of  the  ground  bring  the  locks  very  near  to  each  other,  the  breadth 
of  the  basin  between  them  must  be  increased,  until  it  comply  with 
the  above  condition.  In  general,  this  condition  may  be  attained 
by  increasing  the  space  between  the  locks  ;  and  it  may  be  shown 
by  calculation,  that  in  a  canal,  the  locks  of  which  have  eight  feet 
fall,  there  should  not  be-a  less  space  than  200  yards  between  two 
contiguous  locks. 

In  respect  to  the  loss  of  time  arising  from  the  juxta-position  of 


452  or  CANALS.  [Book  VI 

locks:  In  an  alternating  trade,  but  one  vessel  can  be  in  the  sys- 
tem at  a  time,  while  if  there  be  a  space  between  the  locks,  there 
may  be  an  ascending  vessel  in  each  lock,  and  a  descending  one 
in  each  intermediate  basin.  It  is  therefore  evident  that  if  n  be 

the  number  of  locks  combined  in  a  system,   no  more  than   -th 

part  of  the  number  of  boats  that  would  pass,  were  the  locks  iso- 
lated, can  pass  the  system  in  a  given  time,  when  they  are  in  juxta- 
position.. 

There  are,  however,  local  cases,  in  which  a  system  of  locks, 
in  juxta-position,  must  be  made  use  of,  instead  of  an  equal  num- 
ber of  single  ones  with  basins  between  them.  In  this  case,  the 
defects  we  have  spoken  of,  may  be  obviated  by  making  two  com- 
plete systems  by  the  side  of  each  other.  The  expense  of  con- 
struction is  thus  increased,  but  it  may  be  more  than  compensated 
by  saving  in  other  parts  of  the  work.  Thus,  at  Lockport,  on  the 
great  Western  Canal  of  the  State  of  New-York,  where  the  upper 
level  of  the  canal  lies  on  the  top  of  a  ridge  of  compact  limestone, 
that  descends  rapidly  to  the  lower  level,  it  was  found  much 
cheaper  to  make  two  collateral  systems  of  locks,  than  to  exca- 
vate the  space  necessary  for  intermediate  basins.  In  such  a  com- 
bination, one  of  the  system's  is  used  for  the  ascending,  the  other 
for  the  descending  trade. 

434.  Where  circumstances  compel  the  use  of  deep  locks,  the 
waste  of  water  that  they  occasion  may  be  lessened,  by  making 
lateral  basins  to  receive  a  part  of  the  discharge,  when  a  boat  is 
descending,  and  to  restore  it  to  the  lock  when  a  boat  is  to  ascend. 
One  on  this  plan,  is  described  by  Belidor,  but  the  necessity  of 
such  a  form  can  rarely  occur,  and  the  cost  of  construction  would 
be  very  great. 

435.  The  supply  of  water  that  is  needed  for  a  canal,  depends: 
(l.j  Upon  the  lockage,  which  will  follow  the  law  of  the  num- 
ber of  vessels  that  will  probably  pass.     It  is  usual  to  assume,  that 
a  vessel  will  expend  one  lock  full  of  water  on  each  side  of  the 
summit.     This  will  certainly  be  sufficient  when  the  locks  are  not 
in  juxta-position. 

(2.)  Upon  the  evaporation  from  the  surface,  from  which  must 
be  deducted  the  quantity  of  rain  that  falls  upon  it.  It  has  been 
found  by  experiments  made  upon  the  canal  of  Languedoc,  that 
the  annual  quantity  of  evaporation  is  32  inches.  That  is  to  say, 
that  the  allowance  for  this  cause  of  waste  must  be  equal  to  a  pa- 
rallelepiped of  water,  whose  base  is  the  whole  surface  of  the  water 
in  the  canal,  and  whose  altitude  is  32  inches.  In  most  calcula- 
tions, it  has  been  customary  to  take  this  altitude  at  36  inches. 


Book  VI.]  OF  CANALS,  453" 

(3.)  Upon  the  leakage.  This  may  take  place  through  the 
banks  of  the  canal,  or  through  the  gates  of  its  locks.  In  the 
former  case,  it  will  depend  upon  the  nature  of  the  soil  and  the 
extent  of  the  canal ;  in  the  latter,  it  will  be  a  constant  quantity, 
in  canals  whose'  gates  have  equal  dimensions ;  for  the  leakage 
through  the  gate  nearest  the  summit,  will  supply  that  which  takes 
place  in  all  the  gate  on  the  same  side  of  the  summit. 

The  leakage  through  the  banks  is  greatest  in  new  canals;  for 
the  banks,  if  undisturbed,  are  gradually  tightened  by  their  own 
pressure,  and  by  the  particles  of  earth  which  the  water  deposits 
in  filtering  through  them.  When  the  soil  is  porous,  the  canal 
may  be  lined  with  an  earth  retentive  of  water,  or  a  portion  of  the 
middle  of  each  bank  may  be  built  up  with  a  soil  of  this  character. 
If  placed  in  the  middle  of  the  bank,  a  tough  tenacious^clay  may 
answer  the  purpose.  But  when  upon  the  inner  surface,  it  is  ne- 
cessary that  it  should  as  well  resist  the  action  of  running  water, 
as  the  entrance  of  stagnant.  For  this  latter  purpose,  a  loamy 
earth,  mingled  with  small  pebbles,  has  been  found  to  be  the  best. 
The  operation  of  lining  a  bank  with  earth  retentive  of  moisture, 
is  called  Puddling. 

When  such  precautions  have  been  taken,  and  the  banks  have 
become  consolidated,  it  is  the  estimate  of  European  engineers, 
that  the  leakage  is  twice  as  much  as  the  evaporation,  or  amounts 
to  six  feet  upon  the  surface  of  the  canal,  annually.  In  our  coun- 
try, this  estimate  has  been  found  insufficient :  it  is,  however, 
rather  to  be  ascribed  to  a  defect  in  the  mechanical  construction, 
than  to  any  difference  in  the  physical  circumstances. 

436.  When  a  canal  unites  two  navigable  streams,  and  can  de- 
rive its  waters  from  one  or  both  of  them,  or  when  it  is  parallel  to 
a  considerable  stream,  it  may  be  considered  that  the  supply  of 
water  cannot  fail  to  be  sufficient.  But  when  it  passes  over  a 
ridge  or  summit  that  divides  waters  falling  in  two  directions, 
to  obtain  a  sufficient  supply  will  require  considerable  pains  and 
precautions.  As  the  canal  will  seek  to  traverse  the  ridge  at  its 
lowest  points  or  gaps,  a  channel  cut  from  the  summit  level,  and 
continued  along  the  side  of  the  higher  parts  of  t}ie  ridge,  will  in- 
tercept all  the  waters  that  flow  upon  the  surface  of  these  higher 
parts  ;  and  if  it  have  a  slope  towards  the  canal,  will  convey  them 
to  its  summit  level.  Such  a  channel  is  called  a  Feeder,  and  the 
supply  it  will  furnish  will  be  more  certain  should  it  intersect  the 
course  of  streams.  The  quantity  of  water  it  will  certainly  fur- 
nish, may  be  ascertained  by  gauging  the  streams  :  the  slope  that 
must  be  given  to  it,  will  probably  be  determined  by  local  circum- 
stances ;  and  the  proper  dimensions  to  convey  the  given  quantity 
upon  the  given  slope,  may  be  ascertained  upon  the  principles  of 
§  411. 


454  OF  CANALS.  [Book  VI. 

Even  when  no  stream  of  magnitude  is  traversed  by  the  feeder, 
it  may,  if  the  extent  of  ground  higher  than  it  be  considerable,  re- 
ceive a  sufficient  quantity  of  surface  water  to  feed  a  canal.  This 
may  be  judged  of,  by  knowing  the  extent  of  surface  that  the  feeder 
will  drain,  and  the  quantity  of  rain  that  falls  upon  it.  A  par- 
ticular investigation  will  be  necessary  in  relation  to  the  latter  cir- 
cumstance, for  the  quantity  of  rain  is  far  more  influenced  by  local 
causes,  and  particularly  by  difference  of  altitude,  than  the  quan- 
tity evaporated. 

437.  In  almost  all  climates,  the  quantity  of  water  furnished  by 
streams,  or  directly  by  rain,  is  various  at  different  seasons.     At 
one  time  in  the  year  the  supply  will  be  in  excess,  at  another  in 
defect.     To  guard  against  the  consequences  of  this  inequality, 
reservoirs  must  be  constructed.  These  are  made  by  closing  up  the 
entrances  of  vallies,   into  which  the  feeders  are  conducted,  and 
must  lie  at  so  high  a  level  that  the  water  will  run  from  their  bot- 
tom into  the  canal ;  for,  when  this  is  not  the  case,  no  more  water 
can  be  drawn  from  them,  than  lies  above  the  level,  whence  water 
will  run  to  it.     As  reservoirs  are  liable  to  evaporation  from  their 
surface,  they  ought  to  be  constructed  in  places  that  will  admit  of 
their  containing  a  large  quantity  of  water,  with  the  smallest  pos- 
sible surface  ;  their  banks  ought  therefore  to  be  steep,  and  capa- 
city obtained  by  increasing  their'depth,  rather  than  their  breadth. 
The  water  should  be  drawn  from  their  surface,  in  order  that  it 
may  be  free  from  earthy  matter,  which  the  liquid  remaining  at 
rest  in  the  reservoir,  will  deposit. 

The  strength  of  the  walls  and  banks,  by  which  water  is  re- 
tained in  reservoirs,  follows  the  same  law  as  that  of  the  banks  of 
canals,  §  422,  or  the  walls  of  locks,  §428,  according  to  the  mate- 
rial of  which  they  are  constructed. 

438.  It  is  in  all  cases  of  extreme  importance,   that  none  but 
clear  water  should  be  admitted  into  a  canal.     If  this  precaution  be 
not  observed,  the  canal  will  fill  up,  and  lodgments  will  take  place 
in  the  locks,  that  will  prevent  the  working  of  the  gates.    Hence, 
reservoirs   may   fulfil  an  important  purpose,   by  clarifying  the 
water,  even  when  unnecessary  to  equalize  the  supply.     In  the 
original  execution  of  the  canal  of  Languedoc,  no  precautions  were 
taken  to  admit  none  but  clear  water  ;  on  the  contrary,  efforts  were 
made  to  introduce  into  the  canal,  every  stream  whose  course  it 
intersected.     The  consequence  was,  that  after  a  few  years,  it  was 
under  contemplation  to  abandon  it,  rather  than  incur  the  great 
expenditure  demanded  for  maintaining  it  of  a  depth  sufficient  for 
navigation.  Vauban,  however,  who  was  deputed  for  that  especial 
purpose,  pointed  out  the  means  of  preventing  this  difficulty  from 


Book  VI.]  OF  CANALS.  455 

occurring  again.  For  the  same  reason,  it  is  necessary  in  the 
New- York  canals  to  discharge  the  water  from  them,  and  excavate 
annually  several  inches  of  sediment.  The  banks  are  in  conse- 
quence exposed  to  the  action  of  the  frost,  and  are  rendered  liable 
to  give  way  on  the  re-admission  of  the  water.  These  inconve- 
niences may  be  obviated  by  certain  simple  precautions  : 

(1.)  No  water  should  ever  be  admitted  into  a  canal,  until  it 
has  remained  in  a  reservoir  long  enough  to  discharge  its  impuri- 
ties ;  and  the  water  should,  as  far  as  possible,  be  admitted  at  but 
a  few  points  in  the  higher  levels,  whence  the  rest  are  to  be  sup- 
plied. 

(2.)  When  a  stream  intersects  the  course  of  a  canal,  it  is  not  to 
be  admitted,  but  passed  under  it  by  a  culvert,  or  over  it  by  an 
aqueduct.  <  '-V  • 

(3.)  The  rain,  and  surface  waters  of  the  country,  that  lie 
higher  than  the  canal,  must  be  intercepted  by  the  counter  ditch, 
and  passed  at  proper  places  beneath  the  canal,  to  the  lower 
country. 

439.  When  vtrater  is  to  be  passed   beneath  a  eanal,  a  Culvert 
must  be  constructed.     This  is  an  arched  channel  of  masonry  that 
is  built  beneath  the  bed  of  the  canal,  and  reaches  from  one  side  of 
its  embankment  to  the  other.    If  the  water  to  be  conveyed,  lie  at 
about  the  same  level  with  that  in  the  canal,  the  culvert  has  the  form 
of  an  inverted  syphon,  and  acts  upon  the  principle  of  a  water  pipe. 
The  water  may  then  be  raised  to  nearly  the  same  level  at  the 
end  of  the  culvert,  opposite  to  that  at  which  it  enters,  and  may 
resume  its  former  channel.     When  used  to  convey  water  whose 
level  is  considerably  lower  than  that  in  the  canal,  it  may  be  con- 
sidered as  an  open  channel,  the  arch  performing  no  other  office 
than  to  support  the  embankment  of  the  canal.    The  nrcre  surface 
waters  should  be  conveyed  beneath  the  canal,  at  regular  intervals, 
and  by  small  culverts;  but  streams  will  require  culverts  adapted 
to  their  capacity  ;  and  U*is  generally  better  to  unite  a  number  of 
small  culverts,  than  to  make  the  arched  passage  of  too  large  a 
size. 

440.  When  the  stream  is  large,  or  its  valley  wide,  it  becomes  ne- 
cessary to  convey  the  canal  across  it  by  an  aqueduct.     An  aque- 
duct is  a  bridge,  that  instead  of  carrying  a  road,  contains  the  chan- 
nel of  a  canal  between  its  parapets.     If  built  of  masonry,  it  also 
contains  the  towing  path  and  bank  of  the  canal,  and  a  sufficient 
mass  of  earth  beneath  its  channel,  to  prevent  filtration.     The 
breadth  of  the  channel  is  usually  but  half  as  great  as  in   other 
parts  of  the  canal,  and  therefore  but  one  boat  can  pass  the  aque- 
duct at  a  time. 


456 


OF  CANALS. 


[Book  VI. 


The  following  figure  represents  the  section  of  an  aqueduct  of 
masonry. 


In  our  country,  aqueducts  are  frequently  constructed,  by  form- 
ing a  channel  for  the  canal,  of  plank  confined  by  frames  of  tim- 
ber, as  in  the  figure  beneath.  The  towing  path  is  also  formed  of 
wood. 

In  England,  cast  iron  formed 
into  plates,  and  united  by  screw 
bolts  and  nuts,  is  often  used  for 
aqueducts;  they  may  be  supported 
on  pillars  of  stone  or  iron,  and 
are  perhaps  the  very  best  kind  of 
aqueducts,  except  where  the  cost 
of  the  material  much  exceeds  that 
of  slone  or  wood. 

The  most  remarkable  aqueduct 
of  this  description,  is  upon  the  Ellesmere  Canal,  across  the  valley 
of  Llangollen,  in  Wales. 

The  principles  upon  which  the  dimensions  of  the  several  parts 
of  aqueducts,  may  be  calculated,  are  the  same  with  those  of 
bridges,  and  of  the  banks  of  canals,  or  the  walls  of  locks,  and 
need  not  to  be  repeated. 

441.  Feeders  or  other  sources  of  supply,  may  bring  into  a  ca- 
nal more  water  than  can  be  used  in  lockage,  or  wasted  by  evapo- 
ration and  leakage.  This,  running  over  the  gates  of  the  locks, 
would  cause  a  current  that  must  injure  the  banks,  and  if  the  levels 
be  long,  might  swell  and  overflow  them.  To  prevent  the  water 
accumulating,  Waste  Gates  are  constructed  at  intervals.  These  are 
•  built  of  masory,  and  have  the  shape  of  a  triangular  prism,  the 
edge  of  which  is  at  the  level  beyond  which  the  water  is  not  to 
be  permitted  to  rise.  If  made  in  the  towing  path,  a  bridge  is 
provided  for  the  passage  of  the  horses  employed  in  draught. 

These  waste  gates  are  placed  where  a  natural  channel  has  existed, 
by  the  continuation  of  which  the  surplus  waters  may  be  carried 


.  t 
Book  F/.]  OP  CANALS.  457 

off;  or  where  such  channels  do  not  occur,  an  artificial  channel  is 
formed  for  the  purpose. 

442.  When  a  country  is  mountainous,  and  the  construction  of 
canals  would  be  attended  with  great  and  sudden  changes  of  leve'l, 
locks  become  too  expensive   to   permit  their   application;  and 
it  thus  happens  that  in  many  districts,  where,   canals  might  be 
supplied  with  abundance  of  water,  and  the  wants  of  commerce 
demand  them,  they  are  notwithstanding  considered  impracticable. 
In  such  cases  modifications  of  the  inclined  plane,  rendered  self-act- 
ing, might  no  doubt  be  effected.     But  although  often  proposed, 
and  in  forms  to  which  no  reasonable  objection   can  be  applied, 
they  have  not  yet  been  brought  into  successful  action. 

443.  When  water  is  scarce,  canals  are  impracticable,  or  at  least 
susceptible  of  conveying  only  a  limited  trade.  Some  of  the  modi- 
fications of  the  inclined  plane  use  less  water  than  locks,  and  might 
be  advantageously  employed  in  such  cases. 

Betancourt  has  proposed  a  lock,  which  vessels  may  pass  with- 
out any  expenditure  of  water  whatever.  A  basin  of  area  equal  to 
the  lock  is  built  beside  it,  and  communicates  with  it  at  bottom.  In 
this  basin  is  placed  a  water  tight  case  that  has  nearly  the  same  area 
with  it,  and  is  of  the  same  density  with  water.  A  small  force  will 
therefore  be  sufficient  to  sink  or  raise  it  at  pleasure.  If  the  gates 
of  the  lock  be  closed,  and  this  case  be  depressed,  it  forces  water 
from  the  basin  to  the  lock,  until  it  reach  the  level  of  the  upper 
reach  of  the  canal,  and  a  boat  may  be  thus  raised  up.  If  a  boat  is 
to  descend,  the  gates  are  again  closed  ,  and  the  case  drawn  upwards, 
the  water  in  the  lock  then  flows  back  into  the  lateral  basin,  and 
the  boat  floating  upon  it  -falls  to  the  lower  level.  As  the  depth 
of  water  in  the  basin  varies,  a  consequent  variation  will  be  de- 
manded in  the  force  that  elevates  or  depresses  the  case.  This  is 
effected  by  a  counterpoise,  moving  at  the  extremity  of  the  arm  of 
a  bent  level,  in  an  arc  of  a  circle  whose  plane  is  vertical.  The 
whole  arrangement  may  be  understood  by  the  inspection  of  the 
figure  on  the  succeeding  page. 


5S 


453 


OF    CANALS. 


[Book  VI 


ABODE,  are  the  walls  enclosing  the  chamber  of  the  lock  H, 
and  the  lateral  chamber  I.  F,  is  a  wall  separating  the  chamber  of 
the  lock,  from  the  lateral  basin.  In  this  wall  is  the  arched  opening 
G,  forming  a  communication  between  the  chamber,  H,  and  basin, 
I.  L,  represents  the  upper  gate  of  the  lock  resting  on  the  breast 
wall.  K,a  hollow  plunger,  by  the  elevation  or  depression  of  which 
the  common  level  of  the  water  in  the  basin  and  chamber,  is  raised 
or  lowered,  m  m,  a  chain  passing  over  the  pully,  N,  and  connect- 
ing the  plunger,  K,  with  the  bent-lever,  m  0  P,  at  the  extremity, 
P,  of  which  the  counterpoise  is  situated. 

This  counterpoise  moves  in  the  quadrantal  arc,  r  P  s  ;  and 
when  the  plunger  is  depressed  to  its  greatest  depth,  presses  verti- 
cally on  the  fulcrum,  0,  and  has  no  action  on  the  plunger ;  in  other 
positions  in  the  quadrant,  its  action,  as  has  been  demonstrated  by 
Betancourt,  increases  in  precisely  the  same  ratio  as  that  part  of  the 
weight  of  the  plunger,  which  is  not  supported  by  the  fluid  pres- 
sure of  the  water  in  the  basin. 


T^ook  PL]  PERCUSSION,  &c.  459 

'^KT1  •  •'."*; 

CHAPTER  VIII. 

OP  THE  PERCUSSION  AND  RESISTANCE  OF  FLUIDS. 

s 

444.  When  a  surface  is  exposed  to  the  action  of  a  fluid,  in  mo- 
tion, or  when  a  surface  in  motion  impinges  against  a  fluid,  if  we 
have  no  regard  to  what  occurs  after  impact,  the  circumstances 
may  be  considered  as  identical  ;  that  is  to  say,  the  action  will,  in 
the  one  case,  depend  upon  the  velocity  with  which  the  fluid  strikes 
the  surface;  in  the  other,  on  the  velocity  with  which  the  solid 
strikes  the  fluid.     When  both  are  in  motion,  the  effect  will  ob- 
viously depend  upon  the  difference  between  the  two  velocities, 
or,  what  is  called  the  relative  velocity,  and  either  may  be  con- 
sidered as  being  in  motion  with  this  velocity.     Whichever  of  the 
two  is  in  motion,  or  if  both  be  in  motion,  we  may  consider  the 
action  as  identical  with  a  resistance,  opposed  by  a  plane  at  rest  to 
a  fluid  moving  with  the  relative  velocity  ;  and  the  term  Resist- 
ance may  be,  in  all  cases,  employed  to  denote  the  action. 

The  theoretic  investigation  of  this  problem  is  attended  with 
great  difficulty,  inasmuch  as  the  particles  of  the  fluid  interfere  with 
each  other's  action,  even  before  impact,  and  continue  to  act  after 
impact,  according  to  laws  that  it  is  impossible  to  ascertain. 

445.  If  we  suppose  that  the  fluid  particles  strike  in  guccession 
against  the  surface  exposed  to  them,  losing  by  their  action  so 
much  of  their  velocity  as  is  in  the  direction  of  a  normal  to  the 
surface,  and  that  they  then  cease  to  act,  either  upon  the  surface, 
or  on  the  remaining  particles,  we  have  the  hypothesis  that  is  most 
commonly  employed  in  this  investigation,  and  which  we  shall 
now  make  use  of. 

Let  a  fluid,  whose  density  is  s,  strike  with  a  velocity  r,  at  right 
angles  against  a  plane  surface  whose  area  is  A.  Let  h  be  the 
height  due  to  the  velocity  v,  dx  the  space  through  which  the  fluid 
passes  in  the  time  dt. 

We  have  therefore  from,  (53) 
dx 


The  quantity  of  fluid  that  strikes  against  the  surface  in  the  time 
dt,  will  be  Adx,  and  its  mass 

Aadar;  (478) 

and  as  it  moves  with  the  velocity  u,  its  quantity  of  motion  in  dt 

will  be  • 

Asvdx-,  (479) 


460  PERCUSSION  AND  [Book     VL 

and  in  the  unit  of  time 

A  sv  dx 

-^r-' 

dx 

substituting  the  value  of  -77  from  (477),  we  have  for  the  resist- 
ance R, 

R=A.sv*=2gA.sh.  (481) 

If  the  fluid  be  water,  s=l,  and 

R=At>2=2gA&.  (482) 

In  the  case  of  direct  impact  then,  the  action  of  a  liquid  in  mo- 
tion against  a  surface,  should  be  proportioned  to  the  area  of  the 
surface,  and  the  square  of  the  velocity. 

If  the  fluid  strike  against  the  plane  at  an  angle  of  incidence  t, 
the  velocity  must  be  resolved  into  two  components,  one  of  which 
is  parallel,  the  other  perpendicular  to  the  plane ;  the  latter  will  be 
represented  by  v  sin.  t,  and  the  formula  (482)  becomes 

R=A  (v  sin.  i)2=A  v*  sin.  2i.  (483) 

In  oblique  incidences  then,  the  action  of  the  fluid  should  be 
proportioned  to  the  area  of  the  surface,  the  square  of  the  velocity, 
and  the  square  of  the  sine  of  the  angle  of  incidence. 

To  apply  this  theory  to  cases  that  may  occur  in  practice  : 
Let  the  body  be  a  prism  whose  section  is  an  isosceles  triangle  ; 
let  the  area^of  the  rectangular  ^se  be  A ;  the  angle  of  the  vertex 
of  t^e  triangle  2 « :  The  resistance  on  each  of  the  sides  is 

~  A  v2  sin.  2t, 

• 

and  the  whole  resistance, 

A  «2  sin.  2«.  (484) 

If  the  triangle  be  right-angled,  the  resistance  on  its  faces 
will  bo 

AT)2 

-g-.  (485) 

It  may  also  be  shown,  by  the  application  of  the  calculus  to  the 
same  theory,  that  the  resistance  to  a  half  cylinder,  is  two  thirds 
of  that  to  the  rectangle  which  forms  its  base ;  and  that  the  resist 
ance  to  a  hemisphere  is  one  half  of  that  to  a  plane  surface  of  the 
size  of  its  great  circle. 

446.  These  investigations  being  of  rio  practical  value,  we 
shall  omit  them,  and  proceed  to  the  results  of  experiment,  in  the 
case  to  which  the  hypothesis  bears  the  closest  analogy,  namely, 
that  of  a  jet  of  fluid  striking  against  a  plane  surface. 

From  the  bestiexperiments  that  have  been  made  in  this  case, 
namely,  those  of  Bossut,  it  has  been  concluded  : 


Book   VI.  RESISTANCE  OF  FLUIDS.  461 


(1.)  That  under  similar  circumstances,  the  action  of  the  fluid 
is  nearly  proportioned  to  the  area  of  its  section. 

(2.)  That  all  other  circumstances  being  equal,  the  force  is  nearly 
proportioned  to  the  square  of  the  velocity. 

(3.)  That  the  force  of  the  fluid,  when  the  plane  is  oblicme  to 
the  direction  of  the  current,  does  not  follow  the  law  ol  the 
squares  of  the  sines  of  the  angles  of  inclination.  At  least  at  the 
angle  of  <JO°,  the  resistance  is  always  less  than  would  be  due  to 
that  ratio.  x( 

(4.)  The  absolute  measure  of  the  force  with  which  the  fluid 
strikes  the  plane  directly,  is  not  constant,  or  as  given  in  (482) 

2A  gh, 

but  varies  according  to  the  ratio  that  the  surface  bears  to  the 
section  of  the  vein.  If  the  surface  be  much  greater  than  the 
section  of  the  vein  of  fluid  that  strikes  it,  the  above  formula 
is  true;  but  as  they  approach  more  nearly  to  equality,  the  force 
of  impact  becomes  less,  until,  when  they  are  about  equal,  the 
force  may  be  represented  by 

-Agfc;  (486) 

or  is  no  more  than  three  eighths  of  that  derived  from  the  hypo- 
thesis. 

When  the  phenomena  with  which  these  experiments  are 
attended  are  carefully  observed,  they  are  found  to  be  as  follows: 

The  vein  of  fluid  is  enlarged  as  it  approaches  the  surface,  form- 
ing a  conoid  whose  curvature  is  convex  towards  the  axis,  and 
varies  with  the  greater  or  less  size  of  the  surface.  The  particles, 
after  striking  the  surface,  move  off,  if  the  surface  be  small,  at  a 
great  angle  of  obliquity  ;  and  this  angle  lessens  as  the  surface 
increases,  qntil  they  move  parallel  to  the  surface,  or  as  it  were, 
slide  along  it.  When  this  takes  place,  the  resistance  becomes 
equal  to  that  deduced  from  the  theory,  and  is  no  farther  increased 
by  an  increase  of  the  extent  of  the  surface.  But  if  the  surface 
be  surrounded  by  a  ledge  or  rim,  the  resistance  is  increased  be- 
yond that  deduced  from  the  hypothesis  ;  and  in  a  concave  sur- 
face, the  resistance  which  ought,  according  to  the  hypothesis,  be 
less  than  if  it  were  plane,  is  also  increased  beyond  that  given  by 
the  formula  (482). 


462  PERCUSSION 'AND  [  Book   J7/. 

These  circumstances  may  be  reduced  to  the  test  of  analysis. 


Let  PQ  be  a  surface  exposed  to  the  action  of  a  jet  of  liquid, 
and  let  the  curved  lines,  ALX,  BY  represent  the  boundaries  of  a 
section  of  the  vein  made  by  a  plane  passing  through  the  axis 
MN,  and  let  the  vein  be  a  solid,  generated  by  the  revolution  of 
either  of  these  similar  curves,  an*d  assume  the  surface  acted  upon 
to  have  a  circular  section.  Let  M  be  the  point  where  the  vein  com- 
mences to  spread.  We  may  conceive  that  the  particles  in  motion 
divide  themselves  at  the  point  M,  leaving  within  them  a  conoid  of 
stagnant  fluid  PMQ,  and  that  the  action  upon  the  plane  is  trans- 
mitted to  it  by  this  conoid.  We  may  also  consider  that  the  velocity 
in  the  current  that  surrounds  PMQ,  is  constant,  as  there  is  no  ob- 
vious reason  for  its  change,  and  that  it  is  equal  to  that  the  vein 
had  before  it  began  to  spread. 

This  being  premised,  let  the  area  of  the  vein  at  A  MB  =  a ;  its 
constant  velocity  =  v;  and,  referring  the  curve  to  the  axis  MN,  let 
MG  =  x;  GH  =  g-  the  arc  MH  =s,  and  HL,  the  breadth  of  the 
current  at  H  =z ;  the  radius  of  the  osculating  circle,  at  H  =R ; 
the  angle  NPT  of  the  obliquity  with  which  the  particles  leave  the 
surface  =  <p.  As  the  velocity  is  conceived  to  be  constant,  the 
area  of  the  current  contained  between  the  corresponding  circular 
sections  of  the  outer  and  inner  cone  will  be  constant  also,  and 
this  area  is  formed  at  H,  by  the  revolution  of  the  line  HL, 
which  is  a  normal  to  both  the  curves.  If  HL  be  bisected  in  O, 
and  OF  =  y'  be  drawn  perpendicular  to  the  axis,  this  area  will  be 

2  ir  y'z. 

If  we  take  an  elementary  ring  of  the  moving  fluid,  whose  sec- 
tion is  H  /,  its  mass  will  be 

2  if  g'z  ds ; 
and  its  inner  boundary,  generated  by  the  revolution  of  the  line  H  h, 


Book   VI.]  RESISTANCE  OF  FLUIDS.  463 

Now  by  the  principles  of  §  65,  every  element  of  the  fluid  con- 
tained in  such  a  ring  presses  upon  its  inner  boundary  with  a  cen- 
tral force,  represented  by  (85), 


and  this  force  multiplied  by  the  mass,  2  <  y'z  ds,  will  give  the- 
moving  force  with  which  the  fluid  in  the  elementary  ring  presses 
upon  its  inner  boundary,  2  if  yds.  This  product  is 

2 

T7  .  2iryf  ds.  (487) 

This  pressure  may  be  represented  by  the  weight  of  a  cylinder  of 
the  fluid  that  has  for  the  area  of  its  base  2  if  y  cfo,  and  for  its  alti- 
tude P,  or  by 

2*  g  P  y  ds ; 
and 


whence  we  obtain  for  the  value  of  P, 
v*z      y' 

;  (488) 


and  from  the  general  distinctive  property  of  fluids,  every  point 
of  the  surface,  PQ,  will  be  pressed  by  a  column  whose  alti- 
tude is  P. 

If  now  in  the  preceding  equation  we  substitute,  for  z  and  R, 
their  respective  values, 


d'  d~s 
we  obtain 

dx 
2«gPy  dy=—  A  M  .  j-g  ;  (489) 

and  integrating 

Av2       .  (490) 


dx  dx 

When  2/=0,  we  have  ^  =  1,  and  when  ?/=PN,  we  have  JJ=\/ 

sin.  cp  ;  we  therefore  have  for  the  integral, 

«g  .  P  X  PN2=At>2  (1—  sin.  9)  ;  (491) 

but  the  first  member  of  this  equation  is  obviously  the  value  of  the 
resistance,  for  it  is  the  weight  of  a  cylinder  of  the  fluid  whose  base 
is  the  circle  PN,  and  whose  altitude  is  P  ;  the  resistance,  there- 
fore, is 

Av2  (1—  sin.  9).  (492) 


464  PERCUSSION  AND  [Book  VI. 

It  is  evident  from  this,  that  as  the  angle  <p  decreases,  the  resist- 
ance should  increase  until  <p=90°,  or  when  the  fluid  moves  along 
the  plate  ;  in  this  case  it  becomes 

Av2, 

according  to  the  hypothesis,  and  these  results  are  in  conformity 
with  the  experiments. 

If  a  ledge  be  formed  around  the  surface  PQ,  the  fluid  cannot 
escape  on'the  side  opposite  to  that  on  which  it  impinges,  or  move 
along  the  surface  ;  but  must  be  thrown  off  in  a  direction  making 
the  angle  9  on  the  same  side  of  PQ  with  the  axis  MN  ;  in  this 

dx  dx 

case  -T-  becomes  negative,  and  when  »/=PN,  -7-=  —  sin.  <p  ; 

the  expression  for  the  resistance  therefore  becomes 

Afl2(l-fsin.  <p),  (493) 

and  when  <p=90°, 

2Ar2.  (494) 

We  may  thus  explain  the  very  great  difference  in  the  resistance 
which  a  body  of  the  form  of  a  portion  of  a  hollow  sphere  meets, 
when  moving  in  a  fluid,  with  its  different  surfaces  opposed  to  it.  This 
difference,  it  will  appear  from  the  theory,  may  be  in  a  relation  as 
great  as  four  to  one  ;  the  concave  surface  being  resisted  four  times 
as  much  as  the  convex  ;  and  in  practice,  the  difference  is  even 
greater.  Upon  this  principle  the  parachute  is  applied  to  balloons. 

447.  When  a  surface,  instead  of  being  struck  by  a  vein  of  fluid, 
is  immersed  in  a  mass  of  that  description,  and  has  a  relative  velo- 
city in  respect  to  it,  growing  out  either  of  its  own  motion,  or  that 
of  the  fluid,  or  a  combination  of  both  ;  a  similar  conoid  of  stagnant 
water  will  be  formed  in  front  of  it,  and  a  similar  diminution  of  the 
resistance  will  take  place.  We  can  only  ascertain  the  quantity 
of  this  diminution  by  experiment,  which  shows  that  when  the 
area  of  the  stream  is  great  in  proportion  to  that  of  the  surface, 
the  resistance  may  be  represented  by 


. 

2    :  (495) 

or  is  no  more  than  the  half  of  that  pointed  out  by  the  hypothe- 
sis. 

If,  however,  the  channel  be  narrow,  and  the  surface  fills  up  a 
large  portion  of  it,  the  resistance  augments.  The  experiments 
of  Du  Buathave  given  the  following  results; 

Let  the  resistance  in  a  channel  of  indefinitely  large  size  be 
10000,  and  M  the  ratio  between  the  area  of  the  channel  and  that 
of  the  obstacle,  the  resistance  in  the  channel  will  be 
84600 


Book   P7.]  RESISTANCE  OF  LIQUIDS.  465 

If,  then,  the  obstacle  fill  up  half  the  channel,  the  resistance  is 
more  than  double  what  it  is  in  an  indefinitely  large  channel, 
and  we  may  consider  the  channel  as  large  enough  to  get  nd  of 
this  cause  of  increased  resistance*  when  its  area  exceeds  6|  times 
that  of  the  obstacle. 

448.  When  the  surface  acted  upon,  is  under  the  circumstances 
of  a  floating  body,  one  part  being  immersed  in  the  fluid,  the  other 
floating  above  it,  the  level  of  the  water  is  raised  on  the  side  on 
which  the  fluid  acts,  and  a  depression  takes  place  on  the  opposite 
side. 

The  fluid  striking  against  a  plane  surface  with  the  velocity,  £, 
will  tend  to  rise  in  front  of  it,  to  the  height  due  to  this  velocity,  or 
to  (60) 


In  like  manner,  a  depression  will  take  place  behind  a  solid  body, 
and  if  the  surface  be  a  plane  parallel  to  the  anterior  surface,  the 
whole  difference  of  level  will  be, 

v* 

ff* 

c 

Upon  an  elementary  rectangle,  whose  constant  breadth  =  &,  and 
depth  =  dx,  the  action  will  be  due  in  part  .to  the  velocity  with 
which  it  moves  through  the  fluid,  and  partly  to  a  pressure  due  to 
the  depth,  #,  and  may  therefore  be  represented  (495)  by 

±bdx(v—  V2gx)2.  (497) 

Integrating,  we  have,  for  the  additional  pressure  growing  out  of 
the  rise  of  the  fluid, 


An  equal  resistance  will  grow  out  of  the  depression  behind,  so 
that  the  whole  increase  growing  out  of  this  cause,  will  be 

bv* 
TV-"-  (499) 

O 

It  will  be  obvious,  however,  that  the  fluid  cannot  rise  to  the  height 
due  to  the  whole  velocity,  and  therefore  although  the  resistance 
growing  out  of  this  cause,  probably  varies  with  the  fourth  power  of 
the  velocity,  it  has  a  co-efficient  considerably  less  than  is  deter- 
mined by  the  above  investigation. 

449.  We  may  therefore  infer  that  the  resistance  of  water  to  a 
body  floating  upon  its  surface,  is  composed  of  two  separate  forces, 
the  one  due  to  the  impulse  of  the  fluid  ;  the  other  to  the  wave 
raised  before,  and  the  depression  that  takes  place  behind  it.  The 
first  varies  with  the  square,  the  second  with  the  fourth  power  of 
the  velocities.  The  first,  however,  is  alone  sensible  at  moderate 

59 


466  PERCUSSION,  &c.  [Book  VL 

velocities,  but  the  second  becomes  the  most  intense  at  great  velo- 
cities, and  must  finally  cause  a  limit  beyond  which  the  speed  of  a 
body  moving  at  the  surface  of  a  liquid  cannot  be  carried.  No 
such  resistance  affects  bodies  wholly  immersed  in  a  liquid,  and 
hence,  as  the  limit  of  speed  depends  upon  the  square  of  the  velo- 
city, it  is  not  as  soon  attained  as  in  the  former  case. 

450.  Our  investigations  appear  to  show  that  the  resistance  to 
surfaces  inclined  to  the  direction  at  which  the  fluid  strikes  them, 
varies  with  the  square  of  the  sine  of  the  angle  of  incidence.  This 
is  found  to  be  far  from  true,  in  the  case  of  bodies  moving  in  mas- 
ses of  fluids,  when  submitted  to  the  test  of  experiment.  The 
theory  of  Juan,  makes  this  resistance  to  vary  with  the  sine  sim- 
ply ;  this  is  however  equally,  or  even  more  faulty,  except  when 
the  angle  is  great ;  and  even  in  this  case  it  gives  a  result  below 
the  truth. 

If  the  inclination  of  the  surface  to  the  direction  of  the  fluid,  do 
not  exceed  30°,  or  the  angle  of  incidence  is  between  60°  and  90°, 
the  formula  (483)  corresponds  nearly  with  the  truth;  at  inclina- 
tions from  30°,  to  60°,  the  following  formula  deduced  by  Bossut, 
from  experiment,  will  give  results  nearly  accurate. 

R=f™.2  sin.  fM-0.003  (90°— »)  3-25..  (500) 

At  still  greater  obliquities,  the  following  formula  of  Romme, 
gives  a  nearer  approximation. 

ou3  2+  sin.  «2 

*«  <      R=-r-  •  30  . 


*  „•'•'•? 

Book    VI.]  fUE  MOTION  OF   WAVES.        ,  467 


CHAPTER  IX. 

Or  THE  MOTION  OF  WAVES. 

451.  When  a  pressure  is  applied  to  any  portion  of  the  surface 
of  a  liquid,   the  column  pressed  is  shortened  or  diminished  in 
depth;  the  excess  will  enter  into  the  surrounding  columns,  and 
the  pressure  will,  from  the  nature  of  fluids,  be  propagated  to  them. 
They  will  in  consequence  rise  above  their  original  level.     But 
not    being  sustained  by  a  hydraulic  pressure,    they  will  again 
fall,  and  thus  acquiring  a  velocity  due  to  their  height,  will  de- 
scend below  the  level,  until  that  velocity  is  overcome  by  the  ac- 
tion of  the  adjacent  columns.  These  will,  in  consequence,  receive 
a  similar  motion,  which  they  will  in  turn  propagate.     Thus  a  se- 
ries of  ridges  and  intermediate  cavities,  will  be  formed  upon  the 
surface,  and  appear  to  propagate  themselves  in  circles,  from  the 
column  to  which  the  original  impulse  was  given.       This  motion 
that  appears  to  take  place  on  the  surface,  is  not  a  progressive  mo- 
tion of  the  mass,  for  the  particles  of  liquid  that  pass  from  one  col- 
umn to  another,  return  again  to  that  to  which  they  originally  be-' 
longed  ;  and  even  this  takes  place  below  the  surface.     The  pro- 
pagation of  motion,  therefore,  that  takes  place  at  the  surface,  is 
the  communication  of  a  tendency  to  oscillate,  and  cannot,  so  long 
as  these  oscillations  are  unimpeded,  give  a  progressive  motion  to 
the  particles  at  the  surface. 

452.  The  phenomena  of  the  motions  of  waves  have  been  com- 
pared to  those  of  a  fluid  oscillating  in  a  bent  tube,  and  although 
the  analogy  is  not  complete,  yet  there  are  so  many  points  of  co- 
incidence, as  to  authorize  us  to  adopt  this  phenomenon  as  the  foun- 
dation of  our  theory. 


468 


THE  MOTION 


[Book  VI. 


Let  AB,  CD,  represent  two  branches  of  a  bent  tube,  equally  in- 
clined to  the  horizon.     Let  the  section  of  the  branch  AB— w,  the 


section  of  the  branch  CD=n.  Let  the  vertical  height  EF  of  the 
tube  =p,  and  the  height  of  the  level  of  the  liquid  in  the  branch 
AB,  above  the  common  level  when  at  rest  =  a. 

The  branches  being  equally  inclined,  the  spaces,  in  the  direc- 
tion of  the  axes  of  the  respective  branches,  through  which  the  li- 
quid oscillates,  will  be  proportioned  to  the  vertical  altitude  of  these 
spaces.  Call  the  vertical  height  through  which  the  liquid  oscillates 
in  the  branch  AB,  z.  Then  as  the  liquid  that  is  depressed  in  the 
one  branch,  enters  into  the  other,  the  quantity  that  rises  above  the 
original  level  in  each  will  be  equal ;  the  altitudes  in  the  respective 
tubes,  will  be  inversely  as  their  respective  areas ;  and  the  vertical 
height  of  the  oscillation  in  CD,  will  be 

mz 


453.  To  apply  this  to  the  case  of  waves  diverging  in  circles 
from  a  point.  They  may  be  considered  as  formed  by  the  vibra- 
tions of  liquids,  in  a  series  of  concentric  cylindric  tubes;  the  in- 
tervals between  which  are  infinitely  small.  Their  respective 
areas,  are  therefore  proportioned  to  the  circumferences  of  the  cir- 
cles that  form  the  sections  of  the  cylindric  tubes,  and  as  these  are 
proportioned  to  their  diameters,  the  heights  to  which  a  circular 
system  of  waves  rises,  will  decrease  in  the  inverse  ratio  of  their 
distances  from  the  point  of  original  action,  or  in  an  arithmetic 
progressiou. 

The  force  by  which  the  fluid  is  caused  to  oscillate,  will  be  the 
difference  of  the  pressures  of  the  columns  in  the  two  branches  ; 
this  will  be  measured,  §  331,  by  multiplying  the  area,  m,  by  the 


Book  V[.~\  OF  WAVES.  469 

¥  %  • '  .-'.i"  '  .'&> 

height  a,  supposing  the  density  of  the  liquid  =1,  or  will  be  ma. 
The  column  of  fluid,  whose  pressure  in  the  opposite  branch  is 
equal,  will  be  measured  by  the  product  of  the  area,  w,  into  a  height 
which  is  to  a,  inversely  as  the  respective  areas,  or  by 

amn 

n 

and  the  motion  acquired  by  the  column  in  AB,  will  continue  until 
the  pressure  in  CD  become  equal  to  the  original  force  in  AB,  or 
until  the  column  in  CD  becomes 

2am; 

hence,  the  space  through  which  the  oscillation  in  CD  is  per- 
formed, becomes  equal  to 

2«m 

n 
and 

z=2  a  . 
The  column  in  CD  will  now  preponderate,  and  by  a  similar 

course  of  reasoning  may  be  shown  to  descend  through ,  and 

to  elevate  the  column  in  AB  through  2  a.  Thus,  if  we  abstract 
from  friction,  the  force  exerted  in  each  oscillation  will  remain 
constant ;  the  oscillations  will  therefore  be  constant  in  extent,  and 
the  spaces  described  will  be  proportioned  to  the  forces ;  whence 
we  may  at  once  infer  that  the  oscillations  are  isochronous. 

The  times  of  these  isochronous  oscillations  will  depend  upon 
the  space  through  which  the  motion  is  performed,  and  the  inten- 
sity of  the  moving  force ;  and  as  the-latter  is  equal  in  both  direc- 
tions of  the  oscillation,  the  circumstances  will  be  the  same  as  if 
both  branches  of  the  tube  had  equal  areas. 

Let  the  branches  be  of  equal  diameters,  and  let  m,  be  the  com- 
mon area  of  the  two  branches,  and  a,  the  original  elevation  or  de- 
pression from  the  common  level.  The  difference  between  the 
masses  contained  in  the  two  branches  of  the  tube,  will  be  2  am, 
and  their  sum  will  be  Im ;  /,  being  the  whole  length  of  the  column 
in  both  branches,  or  of  the  axis  of  the  tube.  Applying  the  principle 
of  D'Alembert,  it  may  be  shown  as  in  the  investigation  of  the  prin- 
ciple of  Atwood's  machine,  §  95,  that  the  accelerating  force  at  the 
beginning  of  an  oscillation  is  represented  by  the  difference  of  the 
masses  divided  by  their  sum,  or  by 

2am  _ 

2a 
which  is  equal  to  -y— .     But  this  force  will  be  variable, 

because  the  difference  of  level  in  the  two  branches  of  the  tube  is 
continually  changing.  When  the  liquid  in  one  branch  has  de- 
scended through  the  distance  #,  it  will  have  risen  an  equal  quan- 


THE  MOTION  [Book    VL 

tity  in  the  other  branch,  and  the  accelerating  force/,  will  have  be- 
come 

_    2  a  —  2x 

f=  ~T-; 

and,  as  in  the  formula,  (104a), 

di=^^  gdt  ;  (502) 

and  from  the  general  formula,  (53), 


.  .    .  „., 

substituting  this  value  of  dt,  in  (502),  we  have 
vdv=^  (2adx—2xdx}  . 
Integrating,  and  rejecting  the  constant  quantity,  because  when 


whence 

v=^/  £JL#  (2  ax— a?)  ]  ;  (503) 

substituting  this  value  of  0,  in  the  general  expression,  (53),  we 
have 


dt=- 


whence 


integrating,  and  rejecting  the  arbitrary  constant,  because  when 
x=0,  t=0,  we  have  when  #=a, 


(505) 

i  .  ,1^  .  O 

which  is  the  time  of  half  an  oscillation.     The  time  of  a  complete 
I  -  oscillation  is  therefore 

t=«v(j>-).  ,     (506) 

O 

This  is  by  (286)  the  time  in  which  a  cycloidal  pendulum,  whose 
length  is  half  //would  perform  its  vibrations. 

454.  To  apply  this  to  the  case  of  waves.  A  wave  is  contin- 
ually oscillating  between  its  highest  and  lowest  points,  and  the 
motion  being  considered  analogous  to  that  of  a  fluid  in  a  bent 
tube,  the  time  of  an.  undulation  will  be  that  of  the  oscillation  of  a 


/.]  OF  WAVES.  471 

pendulum,  whose  length  is  half  the  distance  between  the  highest 
and  lowest  points.  The  wave  will  return  to  its  original  height, 
or  will  appear  to  pass  over  the  distance  between  two  contiguous 
elevations  or  depressions  in  twice  this  time,  or  in  the  time  of  a 
single  vibration  of  a  pendulum,  whose  length  is  four  times  as  great 
as  that  whose  oscillations  correspond  with  those  of  the  waves. 
But  if  we  abstract  from  the  elevation  of  the  wave,  the  length  of 
such  a  pendulum  is  the  distance  between  two  contiguous  elevations, 
or  which  is  called  the  Breadth  of  the  wave. 

If  we  call  this  breadth  6,  the  time  T,  of  a  wave  running  its 
breadth,  may  be  represented  by 

T=  «  V  -  =  *  V  —  .  (507a) 

S  S 

Fnom  this  we  obtain  for  the  value  of  the  mean  velocity  with  which 
the  wave  is  propagated,  by  substituting  in  the  general  formula, 


the  value  of  *=2  •_•-/,  and  the  foregoing  value  of  T, 


455.  When  a  series  of  waves  are  proceeding  from  a  centre,  and 
meet  a  vertical  obstacle,  they  are  reflected  ;  for  the  effect  of  the 
obstacle  will  be  the  same  as  if  a  new  impulse  were  given  to  the  col- 
umn, causing  an  elevation  or  depression,  equal  to  that  they  ac- 
tually have  ;  and  this  will  be  propagated  like  the  original  impulse, 
but  in  a  contrary  direction.     The  waves  moving  in  circular  arcs, 
the  reflection  takes  place  in  similar  circular,  arcs  ;  and  thus,  the 
series    of  reflected    waves   will   proceed   exactly  as  if  it  came 
from  a  centre  as  far  distant  behind  the  vertical  obstacle,  as  the 
original  centre  is  in  front  of  it.     If  the  series  of  waves  flow  par- 
allel, and  of  equal  height,  which,  as  we  shall  presently  see,  may  oc- 
cur, the  reflected  waves  will  diminish  in  height,  as  if  they  pro- 
ceeded from  behind  the  obstacle,  and  the  joint  elevation  of  the 
wave  in  immediate  contact  with  the  obstacle  will  be  higher  than 
any  other. 

456.  The  original,  and  reflected  waves,  being  neither  of  them 
attended  with  a  progressive  motion,  will  not  interfere  with  each 
other's  progress  ;   but  where  the  elevations  of  a  wave  of  each  series 
correspond,  the  elevation  is  the  sum  of  the  two  ;  and  where  the 
depressions  correspond,  the  resulting  depression  is  also  the  sum 
of  the  two  depressions.    Where  a  depression  of  one  series  corre- 
sponds with  an  equal  elevation  in  another,  the  surface  of  the  liquid 
will  not  change  its  level.     And  in  two  series  of  circular  waves, 
there  will  be  certain  points  symmetrically  arranged  in  curves, 


472  THE  MOTION  [Book  VI. 

as  might  be  readily  shown  by  a  diagram,  in  which  these  effects 
will  counteract  each  other  ;  and  thus,  although  different  series  of 
waves  do  not  interfere  with  each  other's  propagation,  they  do 
still  neutralize  each  other's  effects,  at  particular  points.  This  ac- 
tion is  styled  Interference. 

457.  If  the  obstacle  against  which  waves  strike,  have  a  vertical 
opening  in  it,  of  small  horizontal  breadth,   the  oscillating  col- 
umns that  reach  it,  will  act  there,  as  an  impulse  originally  ex- 
erted at  that  point  would  have  done  ;  and^  hence,  a  new  series  of 
waves  will  appear  to  proceed  from  the  orifice.     If  the  breadth  of 
the  orifice  be  increased,  new  series  of  waves  will  appear  to  pro- 
ceed from  it,  but  they  will  no  longer  have  the  figure  of  circles, 
for  the  motion  of  oscillation  will  be  propagated  through  the  ori- 
fice, and  act  most  powerfully  in  the  direction  of  a  sector,  whose 
centre  is  at  the  point  whence  the  original  impulse  proceeded. 

458.  When  the  wind  acts  to  raise  waves,  they  do  not  diverge 
from  a  centre,  but  usually  proceed  in  parallel  lines,  straight,  or 
nearly  so.     If  the  impulse  were  momentary,  the  waves  would 
decrease  in  height,  in  consequence  of  the  viscidity  of  the  fluid, 
and  the  friction  among  its  particles.     But  as  winds  blow  for  a 
space  of  time  of  some  duration,  the  original  impulse  is  increased 
rather  than  diminished,  and  thus  waves  continue  to  rise,  and  their 
propagation  may  take  place  with  increased,  rather  than  diminished 
altitude.     The  increase  in  height  will  continue,  until  the  sum  of 
the  columns  elevated  above  the  general   level,  and  the  friction 
become  equal  to  the  disturbing  cause.  The  limit  to  which  a  single 
series  of  waves  can  be  raised  by  the  wind,  has  been  inferred  to  be 
no  more  than  6  feet.  As  wind  blowing  over  the  surface  of  smooth 
water,  moves  parallel  to  it,  the  original  cause  of  waves  being 
raised  by  the  wind,  is  friction  ;  but  after  the  waves  are  raised,  the 
wind  acts  upon  that  surface  which  is  inclined  to  it;  and  its  force 
may  be  resolved  into  two  components,  one  of  which  tends  to  in- 
crease the  elevation.     The  whole  force  of  the  wind  also  tends  to 
give  a  progressive  motion,  to  the  mass  of  water  included  in  the 
elevation  of  the  wave ;  and  thus  the  shape  of  the  waves  ceases  to 
be  a  figure,  with  two  surfaces  equally  inclined  to  the  horizon  ;  and 
the  surface  on  the  side  opposite  to  the  wind,  has  its  inclination 
increased.     This  increase  may  become  so  great  as  to  make  the 
wave  project  beyond  its  base  ;  in  which  case,  the  force  of  gravity 
will  cause  the  summit  to  break,  and  roll  over  the  surface  of  the 
wave  beneath. 

If  a  wind,  after  having  raised  a  series  of  waves,  shall  cease  to 
blow,  and  another  arise  from  the  opposite  point  of  the  compass, 
the  latter  will  act  against  surfaces  more  inclined  to  the  horizon, 
than  the  other  did,  and  will  thus  produce  a  greater  effect.  It  there* 


Book  F/]  or  WATER.  473 

fore  happens,  that  at  sea,  the  highest  waves  are  raised  by  sudden 
changes  of  wind,  or  when  a  wind  blows  in  a  direction  contrary  to 
that  in  which  the  motion  of  oscillation  is  propagated.  Any  change 
in  the  direction  of  the  wind,  will  create  a  new  series  of  waves 
crossing  the  first ;  and  thus  the  elevations  and  depressions,  or  the 
total  height  of  the  waves  may  be  increased.  It  is  in  this  man- 
ner that  the  very  great  excess  of  the  height  of  waves  beyond 
the  limit  stated  for  a  wind  blowing  in  a  constant  direction,  is 
caused.  And  in  conformity,  we  find  the  ocean  comparatively 
smooth  in  those  fegions  of  the  earth  that  are  the  seat  of  constant 
winds,  and  that  the  height  of  waves  is  greatest  in  those  regions 
where  changes  of  wind  are  most  frequent. 

Wind  acting  by  its  friction  to  raise  waves,  it  may  be  inferred 
that  if  any  substance  capable  of  lessening  friction  be  interposed, 
the  elevation  that  would  otherwise  take  place  is  lessened.  And 
in  consequence,  it  has  been  found  that  when  oil  is  poured 
upon  the  surface  of  water,  waves  are  rarely  formed  except  by  the 
most  intense  winds  ;  and  if  poured  upon  waves  already  formed, 
it  permits  the  viscidity  and  friction  of  the  water  to  act  to  bring 
them  to  rest ;  thus,  oil  may  be  used  to  lessen  the  dangers  to 
which  vessels  are  exposed,  by  the  violence  of  the  oscillation  of 
waves,  which  is  in  some  cases  very  great. 

459.  When  waves  meet  ah  inclined  obstacle,  the  columns  in 
which  we  have  conceived  them  to  vibrate,  are  lessened  in  depth, 
and  thus  their  fluid  pressure  is  diminished.  The  waves  no  longer 
meeting  with  the  same  resistance  as  before,  the  liquid  acquires  a 
progressive  motion,  which  will  carry  it  up  the  inclined  surface, 
until  its  moving  force  is  counteracted  by  the  weight  of  the  quan- 
tity thus  elevated  above  its  original  level.  For  this  reason, 
breakers  or  surf,  form  upon  shelving  coasts,  whatever  be  the  di- 
rection of  the  wind. 

When  waves  are  raised  by  the  wind,  the  influence  being  ex- 
erted wholly  upon  the  surface  cannot  penetrate  to  any  great  depth. 
From  30  to  40  feet,  is  inferred  to  be  about  the  greatest  distance 
from  the  surface,  to  which  the  agitation  reaches.  It  is  otherwise 
with  those  waves  that  are  formed  by  the 'attraction  of  the  sun  and 
moon,  and  which  constitute  the  tides. 


60 


474  MOTION  OF  [Book  VL 

CHAPTER  X. 

OF  THE  MOTION  OF  ELASTIC  FLUIDS. 

460.  When  an  elastic  fluid  moves  through  an  aperture  into  a 
vacuum,  it  is  usually  considered  as  contained  in  an  open  vessel, 
through  an  orifice  in  which  it  passes  with  a  velocity  due  to  a  col-, 
umn  of  the  fluid,  of  sufficient  height  to  give  it  by  its  pressure,  the 
density  at  which  it  is  found.  This  hypothesis  is  correct,  so  far 
as  regards  the  equality  of  velocity  between  an  elastic  fluid  con- 
tained in  a  close,  and  in  an  open  vessel ;  provided  their  densities 
be  identical,  for  the  elastic  force  is  by  the  law  of  Mariotte,  §  365, 
exactly  equal  to  the  pressure  by  which  any  given  density  is  pro- 
duced. 

In  consequence  of  this  same  law,  the  density  of  an  elastic  fluid, 
contained  in  an  open  vessel,  will  decrease,  as  we  rise  from  the 
surface  of  the  earth  ;  and  in  order  to  produce  the  usual  existing 
pressure,  the  open  vessel  must  be  considered  as  extending  to  the 
utmost  limits  of  the  atmosphere.  Instead,  however,  of  investiga- 
ting the  circumstances  that  would  actually  take  place,  we  con- 
sider the  elastic  fluid  as  reduced  to  the  liquid  state,  and  as  being 
of  uniform  density  throughout ;  the  height  of  the  column  in  the 
vessel  therefore  becomes  that  which  was,  §  357,  styled  the  height 
of  a  homogeneous  atmosphere. 

If  we  call  this  height,  ft,  we  have  from  §  407, 

u=\/2 gh ; 
and  taking  A=27600  feet,  as  determined  in  §  357,  we  have 

u= 1328  feet.  (508) 

To  take  a  more  exact  determination,  and  which  will  be  applicar 
ble  to  our  succeeding  researches.  . 

At  the  temperature  of  32°,  and  under  a  pressure  of  30  inches 
of  mercury,  the  density  of  that  metal,  in  terms  of  air  as  the  unit, 
is  10467;  hence  the  height  of  a  homogeneous  atmosphere,  of 
the  temperature  of  32°,  is  25268  feet,  and 

u= 1295  feet.  (508a) 

461.  Air,  therefore,  of  the  temperature  of  32°,  will  rush  into 
a  vacuum  with  a  velocity  of  1295  feet  per  second.  If  the  tem- 
perature be  about  60°,  the  velocity  becomes  132S  feet,  at  which 
it  is  usually  stated  in  English  books. 

It  may  be  at  once  inferred  from  this  investigation,  that  when 
the  temperature  of  air  varies,  its  velocity  in  entering  a  vacuum 
will  vary  also. 


Book   VI.~\  ELASTIC  FLUIDS.  475 

In  fact,  if  m  be  the  expansion  of  air  for  each  degree  of  Fahren- 
heit's thermometer  ;  <,  the  number  of  degrees  of  the  thermome- 
ter, reckoned  from  the  freezing  point,  h  becomes  h  (l-p-wif),  and 

t>'=</[2£&(l+mO];  (509) 

whence,  taking  the  above  value  of  1295  feet  for  v,  at  32°,  we  have 
for  «',  at  the  temperature  oH-f-320, 

i/=1295v/(l4-w*).  (510) 

4<52.  When  the  density  of  the  air  varies,  and  we  abstract  the 
variation  of  temperature  with  which  such  variations  are  usually 
attended,  the  height  of  a  homogeneous  atmosphere,  of  the  new 
density,  will  be  the  same  as  before,  and  the  velocity  will  not  vary. 

The  velocity  of  an  elastic  fluid  in  entering  a  vacuum  will,  by 
this  reasoning,  be  always  the  same  with  that  which  a  liquid  of 
similar  density,  and  capable  ofexertingan  equal  pressure  with  it, 
would  flow  from  a  vessel.  And  in  this  form  the  rule  may  be  ex- 
tended to  the  case  of  air,  or  other  elastic  fluids,  rushing  from  a 
vessel  into  a  space  containing  an  elastic"fluid  of  a  different  density. 
The  velocity  will  be  in  all  cases  the  same  as  that  with  which  a  liquid 
of  similar  density,  and  capable  of  exerting  a  pressure  equal  to  the 
difference  of  the  two  tensions,  would  flow.  Thus  air,  of  a  tension 
of  two  atmospheres  having,  if  its  temperature^be  the  same,  double 
the  density  of  that  of  the  atmosphere,  will  flow  out  of  a  vessel  into 
the  open  a*ir,  with  half  the  constant  velocity  at  which  air  would 
enter  a  vacuum. 

When  air  rushes  into  a  vessel  in  which  a  vacuum  has  been 
previously  formed,  its  velocity  is  diminished  as  the  vessel  fills 
with  air,  and  should,  according  to  the  hypothesis  become  =0, 
when- the  air  in  the  vessel  acquires  a  density  equal  to  that  of  the 
air  in  the  space  whence  it  flows.  The  velocity  being  considered 
as  due  to  the  altitude  of  a  homogeneous  atmosphere,  the  motion 
in  this  case  is  considered  as  retarded  by  a  motion  growing  out  of 
a  fall,  through  an  atmosphere  of  equal  and  uniform  density,  by 
whose  pressure  the  density  acquired  at  the  moment  in  the  ves- 
sel, would  be  produced. 

In  this  view  of  the  subject,  the  velocity  with  which  air  rushes 
into  a  close  vessel,  which  it  finally  fills  with  a  mass  of  density 
equal  to  its  own,  is  equably  retarded. 

463.  In  gases  other  than  atmospheric  air,  the  velocities  with 
which  thcry  enter  a  vacuum,  are  in  the  inverse»ratio  of  the  square 
roots  of  their  densities,  for  : 

If  /t,  be  the  height  of  a  homogeneous  atmosphere  of  atmos- 
pheric air ;  /i',  the  height  of  a  homogeneous  atmosphere  of  another 
gas ;  D,  the  density  of  the  gas,  that  of  atmospheric  air  being 


476  MOTION  OF  [Book  VI. 

unity  ;  the  heights,  in  order  to  produce  the  same  pressure,  must 
be  inversely  as  their  densities,  and 

' 


.; 

Hence,  «',  the  velocity  of  a  gas,  whose  density  is  D,  will  be 

*=^-  (511) 

464.  This  theory  is  far  from  being  perfectly  satisfactory,  par- 
ticularly as  it  is,  obvious  that  the  whole  of  the  effects  that  may  be 
due  to  the  elasticity  of  the  air,  are  o.mitted.     The  most  import- 
ant of  these,  perhaps,  is  that  which  takes  place  when  air  rushes 
into  a  vessel,  in  which  a  vacuum  has  previously  been  formed.  In 
this  case  our  theory  would  appear  to  show  that  the  velocity  is 
uniformly  retarded,  until  it  becomes  —0  ;  and  the  density  of  the 
air  that  has  entered  the  vessel  the  same  as  that  without.     This 
does  not  occur  in  practice,  for  the  motion  will  continue  after  the 
densities  become  equal,  and  the  air  in  the  vessel  will  be  con- 
densed ;  it  will  then  re-act  and  expand,  and  the  state  of  rest  will 
be  aquired  by  a  series  of  oscillations. 

465.  It  has  been  ascertained  by  the  experiments  of  D'Aubuisson, 
that  air,  in  passing  through  an  orifice  pierced  in  a  thin  plate,  is 
affected  like  a  liquid,  and'forms  a  vena  contracta,  whose  area  is, 
as  in  the  case  of  a  liquid,  0.62  of  the  area  of  the  orifice.     The 
application  of  cylindric  adjutages,  increases  the  quantity  that  is- 
sues to  0.93,  and  a  conical  tube  to  0.95.     The  adjutage  may  be 
twenty  or  thirty  times  the  diameter  of  the  orifice  in  length,  before 
the  discharge  begins  to  diminish  in  consequence  of  the  friction. 

466.  The  principle1,  §413,  of  the  lateral  communication  of  mo- 
tion, holds  good  in  gases,  as  well  as  in  liquids.     Thus,  liquids  in 
motion  carry  with  them  a  current  of  the  air  that  is  in  contact  with 
them  ;  and  gases,  or  vapours  in  motion,  carry  with   them  the 
neighbouring  air. 

The  latter  fact  may  be  conclusively  established  by  the  phe- 
nomena of  the  Eolipyle.  This  instrument  is  a  boiler  in  which 
steam  is  generated,  and  permitted  to  escape  from  a  narrow  aper- 
ture. It  has  for  ages  been  employed  to  excite  combustion.  Now 
steam  alone,  unmixed  with  atmospheric  air,  wpuld  extinguish 
flame,  instead  of  increasing  its  intensity  ;  and  the  fact  of  its  being 
increased,  proves  that  a  current  of  atmospheric  air  joins  the  efflu- 
ent steam,  and  is  carried  with  it  through  the  burning  fuel. 

We  may  upon  this  principle  explain  a  curious  fact,  observed  in 
the  efflux  of  air  from  a  bellows,  or  other  machine,  in  which  it  is 
compressed  ;  it  has  also  been  observed  in  the  escape  of  steam  from 


;    •  -     «'-v     • 

Book  F/.]  ELASTIC  FLUIDS.  477 

the  safety  valves  of  boilers.  If  a  circular  disk  of  four  or  five 
times  the  diameter  of  the  orifice,  be  placed  close  to  it,  not  only 
will  it  not  be  forced  away  by  the  current  of  elastic  fluid,  but  will 
be  retained  near  the  orifice  by  a  force  of  considerable  intensity, 
in  so  much,  that  if  the  orifice  be  directed  downwards,  the  disk 
will  be  supported  in  spite  of  its  gravity,  even  when  formed  of  a 
dense  metallic  substance. 

That  this  ought  to  be  the  case,  may  be  readily  understood 
when  we  consider,  that  an  elastic  fluid  issuing  from  the  orifice, 
A  B,  and  having  its  course  interrup- 
ted by  the  plate,  G  E  will  assume 
the  form  of  a  conoid,  D  A  B  C,  con- 
taining the  cavity,  G  F  E  ;  this  cavity 
will  at  the  beginning  of  the  action  be 
filled  with  a  conoid  of  ,the  elastic 

fluid.  But  if  a  lateral  communication  of  motion  take  place,  the 
fluid  contained  in  this  conoid  will  join  itself  to  the  stream  that  es- 
capes at  the  edges  of  the  plate,  and  a  vacuum  will  be  formed  in 
the  conoidal  space,  GE  F  ;  the  pressure  of  the  atmosphere  acting 
upon  the  surface  of  the  plate,  G  F,  will  therefore  press  it  towards 
the  orifice.  As  it  approaches  the  orifice,  the  action  of  the  fluid 
will  become  more  intense  ;  for  it  will  strike  against  the  disk ;  the 
surface  beneath  which  the  vacuum  exists, 'will  diminish,  and  thus 
the  force  that  acts  £o  repel  the  disk  from  the 'orifice  may  prepon- 
derate, and  th%  disk  be  forced  back  ;  but  this  force  diminishes 
as  the  disk  recedes,  while  the  surface,  to  which  the  atmospheric 
pressure  is  due,  increases  :  thus  the  forces  that  tend  to  move  the 
disk  in  opposite  directions,  will  be  continually  varying,  and  under 
this  variation  the  disk  will  assume  an  oscillating  motion. 

467.  Air  may  not  only  be  set  in  motion  by  the  difference  in 
pressure,  arising  from  mechanical  expansion,  or  condensation,  but* 
also  by  the  physical  action  of  heat,  which  changes  its  density. 

If  by  any  cause  whatsoever,  the  equilibrium  of  temperature  of  a 
mass  of  air  be  disturbed,  the  parts  which  are  most  heated  become 
less  dense  than  those  which  surround  them,  and  therefore  .tend 
to  rise  ;  the  space  that  they  before  occupied,  will  be  supplied  by 
the  adjacent  air,  and  thus  a  circulating  motion  will  take  place. 

The  force  with  which  a  portion  of  a  mass  of  air  that  has  been 
heated,  will  tend  to  rise,  is  by  the  principle  of  §  334,  equal  to  the 
difference  between  its  own  weight,  and  the  weight  of  an  equal 
mass  of  the  same  air,  before  it  was  heated. 

If  the  air  that  is  heated  be  free,  it  will,  both  in  consequence  of 
the  resistance  it  meets  with  in  rising,  and  the  tendency  of  elastic 
fluids  to  distribute  themselves  over  a  given  surface,  in  such  man- 
ner that  the  pressure  shall  become  uniform,  mix  with  the  air 


478  MOTION  OF  [Book  VI. 

through  which  it  rises ;  it  will  also  assume  a  common  tempera- 
ture with  the  latter,  in  consequence  of  the  radiation  of  its  heat. 

If  the  air  be  heated  in  a  straight  tube,  or  close  channel,  having 
an  aperture  at  both  ends  ;  and  if  the  two  ends  are  not  at  the  same 
level,  it  will  rise  towards  the  upper  end,  and  will  not  mix  with 
other  air,  or  give  out  much  of  its  heat,  until  it  reach  the  higher 
opening.  ^Here  it  will  again  tend  to  mix  and  distribute  itself 
through  the  adjacent  air.  In  this  manner  the  motion  of  air  in 
chimnies  takes  place. 

If  air  be  disseminated  through  a  space  unequally  heated,  and 
its  parts  acquire  the  temperature  of  the  portions  of  that  space 
which  they  occupy,  a  motion  of  circulation  must  also  take  place; 
and  this  will  continue,  so  long  as  the  unequal  distribution  of  tem- 
perature continues. 

In  this  manner,  the  currents  in  the  atmosphere,  called  Winds, 
are  generated,  as  will  hereafter  be  more  fully  explained. 

468.  The  density  of  steam  does  not  vary  exactly  as  its  pressure, 
but  follows  the  specific  law  stated  in  §  373.  Taking  the  relative 
densities  and  temperatures  of  steam,  as  given  in  the  table  of  §  374, 
the  following  results  have  been  obtained. 

TABLE 

Of  the  velocity  with  which  steam  of  different^ensions  enters  a  vacuum. 

Tension  in          I      Velocity  in       I         Tension  in         I       Velocity  in 
Atmospheres.  feet.  Atmospheres.     |  feet. 

1  1910  5  2040 

2  1978  10  2080 

3  2007  15  2122 
^  '••'      4  2023  20  2142 

TABLE 

Of  the  velocity  with  which  steam  of  different  tensions  enters  a  space  con- 
laining  atmospheric  air. 

T.  in  Af.        |        V.  in  Ft.        |        T.  in  At.        |        V.  in  Ft. 

H  874  5  1834 

1J-  1154  8  1952 

2  1400  12  2027 

3  1647  16  2070 

4  1761  20  "2095 


Book  PL]  GASES.  479. 

CHAPTER  XI. 

OF  THE  MOTION  OF  GASES  IN  PIPES. 

469.  When  the  tube  in  which  gases  are  in  motion  is  long,  they 
are  retarded  by  friction,  in  a  manner  analogous  to  that  observed 
in  water  and  other  liquids.  Hence,  although  the  velocity  of  an 
elastic  fluid  cannot  finally  become  constant  in  a  tube,  many  of  the 
circumstances  are  in  other  respects  similar  to  those  stated  in  chap- 
ter IV.  It  is  however  important,  that  we  should  investigate 
them  more  closely  in  the  case  of  air. 

Let  H  be  the  height  of  a  column  of  mercury  that  measures  the 
pressure  of  the  air  on  entering  the  tube  ;  h,  a  similar  quantity  at 
the  place  of  discharge  ;  6,  the  altitude  of  the  barometer  at  the 
time  ;  tf,  the  height  of  the  thermometer  above  or  below  32°  ;  m^, 
the  expansion  of  air  for  each  degree  of  the  thermometer  ;  d,  the 
diameter  of  the  pipe  ;  V,  the  velocity  ;  and  let 


Suppose  the  mean  height  of  the  barometer  to  be  30  inches,  or 
2.5  feet. 

Taking,  as  in  §  459,  the  density  of  mercury,  in  terms  of  air,  at 
32°,  to  be  10467,  the  relation  of  the  density  of  mercury  to  that 
of  the  air  that  is  issuing  from  the  pipe,  will  be 

/2.5T  \ 

10467^)  .  (512) 

The  velocity  t*,  at  the  place  of  discharge,  will  be  (509) 

2.5  T\ 

- 


h.  10467-^;  (513) 

and  extracting  the  square  root  of  the  numbers  under  the  radical 

frj 

0=266  V  (h.       fc)  .  (514) 


The  velocities  at  different  points  in  the  tube  will  be  inversely 
as  the  densities  of  the  air  at  those  points  ;  for  as  the  motion  is 
continuous,  the  same  quantity  of  air  must  pass  through  every  dif- 
ferent section  in  an  equal  time.  The  densities  at  the  ends  being 
proportioned  to  the  pressures,  will  be  proportioned  to  6+H,  and 
6+/i;  the  mean  density  will  be  proportioned  to 

H+fc 
H—2-.  (515) 

And  it  will  be  obvious  that  the  mean  velocity  may  be  obtained  by 
an  analogy,  of  which  the  mean  density  is  the  first  term  ;  the 


430  MOTION  OF  [Book  VI. 

pressure  at  the  place  of  discharge,  the  second ;  and  the  velocity 
•    v,  the  third.     Hence,  we  have  for  the  mean  velocity,  V, 

V=266.  > 


H+/I  (516) 


The  experiments  of  D'Aubuisson  have  shown  that  the  resist- 
ance, of  which  H  —  h  is  the  measure,  is  proportioned  to  the  square 
of  the  velocity.  Other  experiments  show  that  it  is  directly  pro- 
portioned to  the  length  of  the  tube  L,  and  inversely  to  its  diameter. 
Hence,  if  N  be  the  constant  co-efficient  of  the  resistance, 


H—  fc=N.--;  (517) 

and  substituting  in  this  expression  the  value  of  t>  from  (514)  we 
obtain 


whence 

'  '  (519) 


or  in  a  more  convenient  form, 

H 

NLT   -  (520) 

d  (6+/0  ? 

h  being  still  involved  in  the  second  member  of  this  equation,  it 
can  only  be  resolved  by  an  approximation.  This  may  be  ob- 
tained from  knowing  that  in  the  cases  that  most  usually  occur  in 

T 
practice,   ,   .  »   is  a  quantity  that  varies  but  little,  and  which  may, 

therefore,  without  sensible  error,  be  considered  as  constant.  If 
then  we  make 

T 

we  have 

^       H 

(521) 

when  hr  H  Snd  L,  are  estimated  in  English  feet,  and  d  in  inches, 

c=0.002, 
arid  the  formula  becomes 

h=  ~17~  (522) 

: .••*.,.)  0.002  --r+1  . 


Book  VI.]  GASES  IN  PIPES.  481 

The  pressure  under  which  the,  air  issues,  being  thus  obtained, 
the  effluent  velocity  is  given  by  the  formula  (514) 


"=266 
and  here  again  we  may  make  use  of  a  constant  quantity  for 

i  .  ,  ,  without  any  sensible  error. 

To  find  the  quantity  in  cubic  feet,  the  diameter  of  the  pipe  be- 
ing given  in  feet,  the  above  formula  must  be  multiplied  by 
ifd2 


or  if  the  diameter  be  given  in  inches,  by 


4  X  144       576  ' 
and  calling  the  quantity  discharged  Q, 


(523) 

T 

If  we  take,  instead  of  ^  ,  ^,  a  constant  quantity,  determined 

from  experiment  in  the  case  of  the  blowing  machines  of  furnaces, 
we  have,  h  being  estimated  in  feet,  d  in  inches,  and  Q  in  cubic 
feet, 

Q=79.44d2x//i.  (524) 

If  the  area  of  the  aperture,  through  which  the  air  is  discharged, 
should  differ  from  the  mean  area  of  the  tube,  as  is  frequently  the 
case  in  blowing  engines,  the  velocity  may  be  determined  upon 
the  principle,  that  in  different  sections  of  the  same  tube,  the  velo- 
cities are  inversely  as  the  areas,  or,  if  d  be  the  diameter  of  the 
orifice  of  discharge,  D  that  of  the  rest  of  the  tube,  as 
d2 


If  the  orifice  of  discharge  be  in  §,  thin  plate,  this  co-efficient  would 
become 

d2 
0.62  jj^-, 

and  if  formed  by  a  cylindrical  tube, 

d2 
0.93  ^r. 

This  being  the  case  most  usual  in  practice,  we  have  for  the 
quantity  discharged,  from  (524) 

d4 
Q=74.34  jp-v/A.  (525) 

61 


482  MOTION  OF  [Book  VL 


470.  It  will  be  obvious,  from  what  has  been  stated,  that  the 
most  important  application  of  this  subject  is  to  the  tubes,  by 
which  air  is  conveyed  from  blowing  machines  to  excite  the  com- 
bustion of  blast  furnaces  ;  in  these,  a  knowledge  of  the  quantity 
of  air  they  convey,  is  often  of  great  importance. 

Another  useful  practical  case  is,  that  of  the  conveyance  of  in- 
flammable gas  in  pipes,  for  the  purpose  of 'illumination.  The 
above  investigations,  and  the  formulae  thence  deduced,  are  ap- 
plicable in  this  instance  also  ;  for  it  has  been  found  that  the  re- 
sistance to  the  motion  of  carburetted  hydrogen  in  tubes,  not  only 
follows  the  same  laws,  but  is  equal  in  quantity  to  that  which  re- 
tards the  motion  of  atmospheric  air. 

The  same  principles  might  be  applied  to  determine  the  velocity 
of  effluence,  in  terms  of  the  pressure  upon  the  entrance  of  the 
tube.  The  pressure  of  the  issuing  air  may  be  easily  determined 
from  (522)  ;  we  have  not  considered  it  necessary  to  enter  into 
the  investigation  of  formulae  for  this  purpose. 


Book  VL]  AIR  IN  CHIMNIES.  483 

"if.   ' 

CHAPTER  XII. 

OP  THE  MOTION  OP  AIR  IN  CHIMNIES. 

471.  A  chimney  may  be  considered  as  a  tube,  in  any  position 
except  horizontal,  at  the  lower  opening  of  which  air  is  heated  by 
an  extrinsic  cause.  This  air  will,  in  conformity  with  what  has 
been  stated  in  Chapter  X,  rise  and  pass  out  at  the  upper  opening ; 
its  place  will  be  supplied  by  air  pressing  from  the  space  adjacent 
to  the  lower  opening.  If  this  be  heated  in  its  turn,  as  it  enters 
the  chimney,  to  the  same  temperature,  it  will  rise  with  a  force 
equal  to  that  possessed  by  what  preceded  it ;  and  so  soon  as  the 
whole  tube  is  filled  with  the  heated  air,  the  velocity  will  become 
uniform. 

The  circumstances  will  obviously  be  the  same  as  if  a  tube  were 
adapted  to  the  bottom  of  the  chimney  and  filled  with  air  of  the 
original  temperature,  while  the  chimney  is  filled  with  the  heated 
air  ;  and  thus  the  air  will  move  as  in  an  inverted  syphon,  in 
which  two  columns  of  fluid,  of  different  densities  but  of  equal 
altitudes,  press  against  each  other.  For  as  the  action  of  the  ex- 
ternal air  may  be  considered  equal  on  both  openings  of  the  chim- 
ney, the  acceleration  it  produces  in  the  one  column,  and  the  re- 
tardation in  the  other,  may  be  neglected. 

Let  A,  be  the  height  of  the  column  of  air  in  the  chimney  ;  t  the 
external  temperature  ;  ?  that  of  the  air  in  the  chimney ;  m  the 
dilatation  of  the  air  for  each  degree  of  the  thermometer.  The 
length  of  the  column  of  cold  air  reduced  to  32°,  will  be 

1  A 


l+W 
The  same  column  at  the  temperature  t'9  will  be  (509), 


•>'  -"•  '.  v>.       (527) 

and  this  will  be  the  height  to  which  the  velocity  would  be  due, 
with  which  the  air  would  enter  the  chimney,  if  that  were  void  of 
air ;  the  actual  velocity  will  be  due  to  the  difference  between  this 
height  and  ft,  or  to 

" 


and  we  have  for  the  value  of  v  from  (509) 

«=avF:i»ffi(S^-l)],  (529) 


484  MOTION  OF  [Book  VI. 


the  quantity  1-f  tm,  is,  generally  speaking,  so  small  that  it  may 
be  neglected  ;  and  we,  therefore,  have 

v  =  7  [Zghm  (*—*')]  ,  (531) 

and  we  may  take  for  the  height  to  which  the  velocity  is  due, 
hm(t—t').      . 

472.  Such  would  be  the  theory  were  the  air  to  meet  with  no  re- 
sistance in  its  passage  through  the  chimney.  But  it  is  obvious  that 
it  will  meet  with  friction  ;  that  its  temperature  and  consequent 
ascensive  force  will  diminish.  Thus  the  velocity  with  which 
the  heated  air  would  otherwise  begin  to  ascend  will  be  lessened  ; 
and  its  excess  of  elastic  force  will  continually  decrease,  from  the 
origin  to  the  summit.  At  the  extremity  of  the  chimney,  the 
elastic  force  will  obviously  be  proportioned  to  the^xcess  of  temper- 
ature that  it  retains  at  that  point.  And  as  we  may  consider  that 
the  same  quantity  of  air  passes  through  every  different  section 
of  the  chimney  in  the  same  time,  it  follows  that  at  every  point 
the  velocity  is  inversely  as  the  densit}^  and  therefore  that  the 
velocity  decreases  from  the  origin  to  the  summit. 

If  P  be  the  pressure  under  which  the  air  enters  the  chimney  ; 
p  the  pressure  it  retains  at  the  summit.  The  loss  of  motion 
growing  out  of  resistances  in  the  chimney,  may  be  represented 
byP—  p. 

The  resistance  appears  from  experiment  to  be  directly  pro- 
portioned to  the  square  of  the  velocity,  and  the  length  of  the  chim- 
ney ;  and  inversely  to  its  diameter. 

If  p  be  estimated  as  the  height  of  a  column  of  the  heated  air, 
under  the  pressure  of  the  atmosphere, 

p=hm(t—t'}',  (532) 

and  from  (509)  and  (531), 

v=  v/2gft,  (533) 

calling  the  velocity  V  ;  the  diameter  of  the  tube  D  ;  its  length 
L  ;  and  the  co-efficient  of  the  resistance  K  ;  the  law  of  the  resist- 
ance just  stated,  gives  us 

V2L 


whence 

KV2L 


substituting  this  value  in  the  foregoing  equation,  (533),  we  have 

K  V2  L 

(534) 


Book  VI.']  AIR  IN  CHIMNIES.  485 

whence  we  obtain  for  the  value  of  V, 

/       PD       \ 

V=^^(]5+2iIx)'  (535) 

and  for  that  of  K, 


_—  V2D 


' 


By  means  of  which  last  formula,  the  co-efficient  of  the  friction 
can  be  obtained  by  experiment.  This  has  been  done  by  Peclet, 
who  has  found  when  the  values  of  P,  D,  L,  and  V,  are  given  in 
French  metres  : 

(1.)  That  in  brick  chimnies, 

K=0.0127.  (537) 

(2.)  In  wrought  iron  flues, 

K=0.0050.  (538) 

(3.)  In  chimnies  or  flues  of  cast  iron,  •>  :,"   . 

K=0.0025.  (539) 

473.  The  great  difference  between  the  values  of  the  co-effi- 
cient of  the  friction,  in  tubes  of  different  substances,  is  a  remark- 
able fact;  particularly  as  it  differs  essentially  from  what  is  ob- 
served in  the  motion  of  water  in  pipes,  where  the  substance  of 
which  they  are  composed  has  no  essential  influence  on  the  ve- 
locity. 

This  discrepancy  may  be  explained  by  the  difference  in  the 
attraction  of  the  two  fluids  for  the  substances  of  which  the  tubes 
are  composed.  Water  adheres,  by  its  attraction  of  cohesion,  to 
the  pipes,  and  moistens  them.  It  thus  in  fact,  in  running  in 
tubes,  rubs  against  the  water  that  adheres  to  them,  and  the  fric- 
tion is  a  constant  quantity  ;  for  it  always  takes  place  between 
surfaces,  both  of  which  are  composed  of  the  same  substance. 
While  in  the  case  of  the  motion  of  air,  the  friction  takes  place 
between  it  and  the  material  of  which  the  tube  is  composed,  and 
its  co-efficient  should  therefore  be  different  in  tubes  of  different 
materials. 

The  dimensions  of  chimnies  depend  upon  the  quantity  of  air 
which  the  combustible  requires  for  /its  perfect  ignition,  and  upon 
the  velocity  which  it  will  assume  in  them.  The  former  i*an 
object  of  chemical  and  physical  investigation,  and  would  be  out 
of  place  in  a  work  on  Mechanic^.  The  latter  is  determined  by 
the  foregoing  equations. 


486  THE  WINDS.  [nook 


CHAPTER  VIII. 

OF  THE  WINDS. 

474.  Winds  are  currents  in  the  atmosphere,  that  are  generally 
if  not  always  caused  by  a  disturbance  in  its  equilibrium,  owing 
to  the  unequal  and  variable  distribution  of  temperature  upon 
the  surface. 

The  temperature  of  the  surface  of  the  earth,  is  due  to  two  an- 
tagonist causes : 

(1).  The  constitution  of  the  mass  of  the  earth  and  its  atmos- 
phere : 

(2).  The  reception  of  the  radiant  heat  of  the  sun. 

The  heat  arising  from  the  first  of  these  causes  radiates,  and  in 
consequence  of  this  radiation,  the  earth  would  continually  grow 
cooler.  This  waste  is  supplied  by  the  heat  that  radiates  from 
the  sun.  It  has  been  proved  incontestibly  by  Laplace,  that  the 
mean  temperature  of  the  earth  is  now  constant,  and  has  been  so 
for  2000  years  ;  hence,  the  quantity  of  heat  that  radiates  from  the 
earth,  and  the  quantity  received  from  the  sun,  exactly  balance 
each  other:  that  is  to  say,  for  any  Aeries  of  years,  the  sum  of  the 
quantities  received  by  the  whole  earth  from  the  sun,  is  just  equal 
to  the  quantity  that  radiates. 

This  equality  does  not  exist  for  short  periods  of  time.  The 
radiation  from  the  surface  goes  on  continually,  although  not 
uniformly,  being  greatest  from  the  portions  that  are  most 
heated  ;  while  the  reception  of  heat  at  aoy  given  place,  only  takes 
place  while  the  sun  is  above  the  horizon. 

The  quantity  of  heat  received  from. the  sun,  upon  a  given  sur- 
face, varies  in  a  given  natural  day  with  the  altitude  of  the  sun 
above  the  horizon,  in  consequence  of  the  greater  or  less  obliquity 
of  his  rays.  It  also  differs  from  day  to  day,  in  consequence  of 
the  variation  in  the  length  of  the  natural  day,  and  in  the  meri- 
di^i  altitude  of  the  sun. 

The  first  of  these  variations  grows  out. of  the  rotation  of  the 
earth  upon  its  axes,  in  the  space  of  a  day  ;  the  second,  arises  from 
the  revolution  of  the  earth  around  the  sun,  in  an  annual  orbit 
inclined  to  the  equator. 

This  orbit  being  an  ellipse,  of  which  the  sun  occupies  one  of 
the  foci,  and  the  arcs  of  the  ellipse  described  not  being  propor- 
tioned to  the  times,  the  sun's  daily  apparent  motion  is  not  equa- 
ble. 


Book  VL~\  THE  WINDS.  487 

The  orbit  being  inclined  to  the  axis  of  rotation,  causes  a  vari- 
ation in  the  length  of  the  natural  day,  and  in  the  sun's  meridian 
altitude;  hence  flow  the  vicissitudes  of  the  seasons.  The  diame- 
ter of  the  orbit  that  passes  through  the  sun,  does  not  pass  through 
the  equinoctial  points,  but  is  nearer  to  the  solsticial  points  than 
to  the  equinoctial ;  hence  the  equinoxes  do  not  divide  the  year 
into  two  equal  parts,  but  the  summer  of  the  northern  hemisphere 
is  about  7|  days  longer  than  the  summer  of  the  southern.  It  fol- 
lows that  the  northern  hemisphere  receives  more  heat  from  the 
sun,  than  the  southern;  and  a  balance  taking  place  in  the  quantity 
of  heat  received  and  radiated,  not  only  in  the  whole  earth,  but  in 
the  two  separate  hemispheres,  the  northern  hemisphere  is 
warmer  than  the  southern  ;  therefore,  even  the  mean  place  of  the 
equator  of  temperature  does  not  coincide  with  the  astronomic 
equator,  but  lies  north  of  it.  -^ . 

Did  the  plane  of  the  earth's  orbit  coincide  with  the  terrestrial 
equator,  the  days  and  nights  would  be  constantly  equal  in  every  part 
of  the  globe.  The  sun's  rays  at  noon  would  fall  vertically  upon 
points  in  the  equator,  and  would  be  tangents  to  the  earth  at  the 
poles ;  and  were  there  no  lateral  communication  by  radiation  or 
the  conducting  property  of  the  materials  of  the  earth,  and  its  at- 
mosphere, the  quantity  of  heat  received  at  noon,  from  the  sun, 
at  every  different  place,  would  obviously  be  proportioned  to  the 
cosine  of  the  latitude.  The  mean  temperature  of  the  days  in 
every  latitude  would  be  constant;  and  this  mean  diurnal  tem- 
perature would  follow  a  regular  law  of  decrease,  from  the  equator 
to  the  poles.  Taken  for  a  whole  year,  the  mean  temperature,  if 
disturbing  causes  did  not  act,  should  follow  a  similar  law.  The 
variations  at  the  surface  are  so  sudden  as  to  cloak  this  law,  except 
when  studied  by  the  comparison  of  a  long  series  of  thermometric 
observations ;  but  when  we  penetrate  beneath  the  surface  of  the 
earth,  to  a  depth  sufficient  to  be  removed  from  the  influence  of 
the  changes  at  the  surface,  we  find  this  law  to  hold  good  :  >  fV 

The  temperature  at  the  depth  of  from  60,  to  SO  feet  beneath 
the  surface  of  the  earth,  is  constant  in  every  different  climate,  and 
corresponds  closely  with  the  mean  temperature  at  the  surface. 

At  the  equator,  the  days  and  nights  are  of  equal  lengths  through- 
out the  year  ;  and  the  meridian  zenith  distance  of  the  sun,  never 
amounts  to  more  than  231°.  At  the  polar  circle,  the  natural  days 
vary  in  length  from  24  hrs.  to  0  hrs.,  and  the  change  in  the  me- 
ridian altitude  is  from  0°  to  47°.  Thus,  at  the  equator,  the  tem- 
perature varies  but  little  on  each  side  of  the  mean  ;  while,  were 
the  earth  of  uniform  surface,  the  extent  of  the  variation  on  each 
side  of  the  mean  rate  should  increase  regularly  from  the  equator, 
to  the  polar  circles.  Within  the  polar  circles,  the  sun  does  not 


488  THE  WINDS.  [Book  VI. 

rise  for  several  days  on  each  side  of  the  winter  solstice,  and  is 
above  the  horizon  for  several  days  near  the  summer  solstice; 
while  at  the  poles,  the  natural  day  and  night  have  each  a  length 
of  six  months.  Hence  it  might  be  inferred,  that  even  greater 
variations  on  each  side  of  the  mean  temperature  should  occur 
within  the  polar  circles,  and  that  the  variation  should  be  a  maxi- 
mum at  the  pole. 

The  quantity  of  heat  derived  from  the  sun  on  the  clay  of  the 
summer  solstice,  has  been  calculated  to  be  nearly  equal  in  the 
lats.  of  45°  and  60°  ;  and  when  the  sun's  declination  exceeds  18°, 
the  quantity  of  heat  received  in  24  hours  at  the  pole  is  not  less 
than  it  is  at  the  equator,  during  the  twelve  hours  the  sun  is  above 
the  horizon. 

It  thus  happens  that  were  there  no  disturbing  cause,  nor  any 
means  by  which  the  excess  of  heat  might  be  conveyed  from  one  re- 
gion to  another,'  the  distribution  of  heat  at  the  surface  would  be 
continually  varying.  The  distribution  according  to  a  regular  law 
of  decrease,  from  the  equator  to  the  poles,  would  only  take  place 
near  the  time  of  the  equinoxes  ;  while  at  the  solstices  the  parallels 
receiving  the  greatest  quantity  of  heat  would  be  without  the 
tropics ;  and  parallels  in  the  frigid  and  torrid  zones  might  re- 
ceive equal  quantities  of  heat  on  the  same  day. 

As  however  the  earth  has  at  no  great  depth,  the  mean  tem- 
perature of  the  climate,  this  will  tend  in  a  high  latitude  to  prevent 
the  surface  from  acquiring  a  heat  as  great  as  that  actually  com- 
municated on  the  hottest  days  ;  and  thus  the  heat  of  the  surface 
in  such  latitudes  will  never  rise  as  high  on  the  warmest  days,  as 
is  consistent  with  the  quantity  actually  received.  In  the  same 
manner  the  surface  of  high  latitudes  never  cools  as  low  as  is  con- 
sistent with  the  quantity  of  radiation  from  the  surface,  in  the  ab- 
sence of  any  supply  from  the  sun. 

Local  circumstances  that  will  hereafter  be  stated,  affect  the 
range  of  sensible  heat ;  and  thus,  places  in  the  same  latitude,  many 
have  very  different  maxima  and  minima  of  temperature;  and  the 
amount  of  variation  may  be  much  greater  in  a  given  place,  than 
it  is  in  another  of  the  same  latitude. 

Certain  physical  causes  interfere  in  high  latitudes,  to  prevent 
the  extent  of  the  changes  of  temperature  being  as  great  as  they 
would  be,  in  consequence  of  the  great  difference  in  the  altitude  of 
the  sun,  and  of  its  continuance  above  the  horizon  at  different  sea- 
sons ;  these  will  be  stated  in  their  proper  place;  and  thus  the 
greatest  alternations  seem  to  take  place,  in  thelat.  of  from  35°  to 
50°.  In  New-York,  the  annual  range  of  the  thermometer,  from 
its  summer  maximum  to  its  winter  minimum,  sometimes  exceeds 
"*"*°  ;  and  the  difference  between  the  mean  temperature  of  the 


Booh  VI.]  THE  WINDS.  489 

hottest,  and  that  of  the  coldest  month,  amounts  to  56°  ;  at  Pekin, 
the  latter  difference  is  60°,  while  atFunchal,  it  is  no  more  than  10°. 

475.  Difference  of  elevation  above  the  surface  of  the  earth,  has 
a  great  effect  upon  the  temperature  of  places.  The  air  of  the  at- 
mosphere is  from  its  elastic  nature,  denser  at  the  mean  surface  of 
the  earth,  than  in  higher  regions  ;  and  air  has  an  increased  capa- 
city for  heat  when  it  becomes  rarer  ;  hence,  in  the  higher  parts 
of  the  atmosphere  an  intense  cold  prevails,  and  the  temperature 
of  the  land  decreases  with  its  elevation  above  the  level  of  the  ocean. 
So  intense  is  the  action  of  this  cause,  and  so  speedily  is  it  sensible 
in  rising  from  the  earth,  that  even  in  the  heart  of  the  torrid  zone 
mountains  exist  whose  tops  are  covered  with  perpetual  snow.  On 
these,  it  is  therefore  evident,  that  the  thermometer  never  rises 
much  above  32°. 

It  has  been  inferred,  but  without  sufficient  reason,  that  the 
mean  temperature  of  the  limit  of  perpetual  snow  is  32°,  but  ob- 
servation shows  that  it  is  in  all  cases  lower,  and  the  limit  appears 
to  arise  rather  from  the  mean  temperature  of  the  warmest  month, 
than  from  that  of  the  entire  year. 

It  has  been  estimated  that  the  temperature  decreases  as  we  re- 
cede from  the  surface  of  the  earth,  at  the  rate  of  about  1°  for 
every  270  feet. 

In  consequence  of  the  great  variation  that  takes  place  in  the 
quantity  of  heat  received  from  the  sun,  in  temperate  and  frigid 
climates,  while  the  radiation  has  a  much  less  range,  an  accumula- 
tion takes  place,  at  those  seasons  when  the  sun  is  highest  at 
noon,  and  remains  longest  above  the  horizon,  by  which  the  tem- 
perature increases  for  some  time  after  the  solstice  ;  a  correspond- 
ing diminution  in  temperature  goes  on  after  the  shortest  day.  Thus 
it  happens  that  the  greatest  heat  in  middle  latitudes  occurs  about 
a  month  after  the  summer  solstice,  and  the  greatest  cold  about  an 
equal  time  after  the  winter  solstice. 

An  empirical  formula,  that  very  nearly  corresponds  with  ob- 
servation, has  been  framed  to  represent  the  temperatures  at  dif- 
ferent seasons,  and  at  altitudes  above  the  level  of  the  sea,  in  all 
latitudes. 

Let  M  be  the  mean  temperature  in  lat.  45°  ; 
M+E,  the  mean  temperature  at  the  equator  ; 
L,  the  latitude  of  the  place  ; 
F,  a  co-efficient  determined  by  observation  ; 
H,  the  altitude  of  the  place  above  the  level  of  the  sea  ; 
/,  the  sun's  longitude. 

Then  we  have  for  the  mean  diurnal  temperature,  on  the  day 
for  which  the  longitude  /  is  given, 

.  2  L+F  sin.  (I— 30°)—  ^  .       (540) 
62 


490  THE  WINDS.  [Book    VI. 

If  F— 15°,  the  formula  gives  results  that  are  on  the  average 
true,  in  the  western  part  of  Europe,  and  in  the  North  Atlantic. 

476.  The  nature  of  the  surface  has  a  great  effect  upon  the  dis- 
tribution of  temperature,  and  upon  the  distance  that  exists  between 
the  extremes  of  heat  and  cold  in  different  parts  of  the  globe. 

The  surface  of  the  earth  is  partly  of  solid  land,  and  partly  wa- 
ter. Within  the  former,  the  communication  of  heat  is  extremely 
slow,  and  hence  the  surface  of  the  land  adapts  its  temperature  more 
closely  to  the  quantity  of  heat  received  daily,  than  the  surface  of 
the  ocean.  The  latter,  when  exposed  to  heat  that  varies  from  place 
to  place,  is  set  in  motion  ;  for  so  long  as  the  temperature  of  the 
surface  does  not  fall  below  40°,  the  water  expands,  and  the  columns 
in  the  warmer  parts  increasing  in  altitude,  while  they  diminish  in 
density,  a  current  is  caused  from  the  parts  most  heated,  to  those 
which  are  colder;  a  counter  current  is  formed  in  the  water  be- 
neath, in  which  the  colder  portions  flow  towards  the  zone  of 
greatest  heat.  In  this  manner,  so  much  of  the  heat  derived  from 
the  sun,  as  exceeds  the  radiation,  is  conveyed  at  the  surface,  from 
the  heated  regions,  to  those  which  are  colder. 

This  motion  ceases,  however,  when  the  temperature  of  the 
surface  falls  below  40°,  beneath  which  degree  any  diminution  of 
the  temperature  of  the  water  will  render  it  lighter  than  that  which 
is  beneath,  and  the  heated  portion  sinks,  instead  of  rising. 

When  the  sun  shines  upon  the  land,  its  calorific  rays  penetrate 
to  but  a  small  depth,  say  no  more  than  a  few  inches ;  its  surface 
is  in  consequence  rapidly  heated,  when  the  heat  received  exceeds 
that  which  is  radiated  :  when  the  latter  is  in  excess,  the  loss  of 
heat  is  principally  confined  to  the  surface,  which  is  therefore 
rapidly  cooled. 

In  water,  when  the  reception  of  heat  exceeds  the  radiation,  the 
calorific  rays  penetrate  to  a  considerable  depth,  say  20  to  30  feet ; 
the  heat  being  thus  distributed  through  a  large  mass,  the  superfi- 
cial temperature  is  but  slowly  altered.  When  on  the  other  hand, 
the  radiation  is  in  excess,  the  upper  portions,  on  parting  with  their 
heat,  contract,  and  becoming  heavier  than  the  water  which  is  be- 
neath, descend  until  they  reach  the  bottom,  or  a  stratum  of  the 
fluid  of  equal  temperature  with  themselves;  a  circulation  is  thus 
kept  up,  and  the  heat  lost,  although  equal,  or  even  superior  in 
quantity  to  that  withdrawn  from  the  land,  is  again  derived  from 
a  large  mass  ;  the  diminution  of  the  superficial  temperature  is 
therefore  slow.  When  however  the  surface  is  qpoled  below  40°, 
this  motion  ceases. 

From  the  combination  of  these  circumstances,  it  happens  that 
the  temperature  of  the  surface  of  the  ocean  is  more  constant  than 
that  of  the  land  ;  that  it  can  be  reduced  to  certain  laws  easily  dis- 


Book  VL]  THE  WINDS.  491 

covered  from  observation  ;  and  that  it  follows  much  more  closely 
than  the  land,  the  law  of  a  regular  diminution  of  temperature, 
from  the  equator  to  the  poles.  The  variations  on  each  side  of  the 
mean  temperature,  are  also  less  on  the  ocean  than  they  are  upon 
the  land.  These  rules  likewise  hold  good  in  islands,  and  to  a  less 
extent  in  countries  adjacent  to  the  ocean ;  these  portions  of  the 
land  have  climates  of  less  vicissitude  than  the  interior  of  conti- 
nents. 

477.  Great  and  deep  lakes  have  a  similar,  although  less  impor- 
tant influence  on  climate  ;  for  although  the  extent  of  their  sur- 
face be  not  sufficiently  great  to  cause  any  distribution  of  heat  by 
currents,  the  difference  between  the  quantities  of  heat  received 
and  radiated,  affect  not  their  surface  alone,  but  their  whole  mass. 
Their  surface,  therefore,  like  that  of  the  ocean,  preserves  a  more 
uniform  temperature  than  that  of  the  land. 

When  a    lake  cools,  the  motion  that  we  have  described  in 
speaking  of  the  ocean,  in  which  the  cooler  parts  descend,  and  by 
which  the  heat  is  withdrawn  from  the  whole  mass,  goes  on  until 
the  temperature  throughout  becomes  40°.     Water  at  this  tem- 
perature reaches  its  maximum  of  density,  the  motion  of  descent 
ceases,  and  the  surface  will  be  speedily  cooled  to  the  temperature/ 
of  congelation.     Deep  lakes,  however,  descend  to  such  depths/^ 
to  come  into  contact  with  those  strata  of  the  earth's  mass 
tain  the  mean  temperature  of  the  climate  ;    from  these  tj 
will  derive  heat;  and  thus  it  may  happen  that  a  deep>^e>  °f  no 
great  superficial  extent,  is  never  frozen.     Such  phep^"ena  occ"r 
in  the  small  lakes  of  the  western  part  of  the  stat^"  New- York, 
the  surface  of  which  never  freezes. 

Shallow  lakes  and  morasses  tend  to  makeX climate  colder ;  for 
the  cold  produced  at  their  surfaces  not  onty  by  evaporation,  but 
by  radiation,  cannot  long  be  compensatory  an  internal  motion. 

The  draining  of  morasses,  renders  A  climate  warmer,  as  does 
the  cutting  of  forests,  and  the  extern  of  cultivation.  The  ef- 
fects of  the  latter  causes  appear  to/xtend  beyond  the  region  where 
they  operate  directly.  Thus,  t*e  cultivation  of  France  and  Ger- 
many, has  changed  the  climate  of  Italy  ;  and  thus,  the  clearing  of 
the  forests  of  the  interior  of  the  United  States,  has  raised  the 
mean  annual  temperature  of  the  seacoast. 
478.  To  recapitulate  our  general  inferences  : 

(1).  Upon  the  land,  the  zone  of  greatest  sensible  heat  will  be 
a  little  north  of  the  equator  on  the  days  of  the  equinox;  but 
will  on  other  days  of  the  year  vary  in  position  ;  and  will  be  found 
in  the  interior  of  continents  about  a  month  after  the  solstices, 
in  latitudes  as  high  as  from  40°  to  50°.  In  the  ocean,  on  the 


492  THE  WINDS.  [Book   VI. 

contrary,  the  zone  of  maximum  temperature  does  not  vary  in  its 
position  more  than  8°,  and  is  always  to  the  north  of  the  equator. 
From  this  zone,  the  heat  of  the  surface  of  the  ocean  decreases 
uniformly  to  a  latitude  of  from  28°  to  30°.  Beyond  this  limit,  on 
either  side  of  the  equator  to  the  latitude  of  50°,  the  heat  of  the 
surface  is  alternately  greater  or  less  than  would  be  consistent 
with  a  regular  decrease,  according  to  the  law  of  the  cosine  of  the 
latitude;  but  after  the  equinoxes,  it  appears  to  coincide  for  a 
short  time  with  the  results  of  that  law. 

(2).  Elevated  countries  are  colder  than  those  more  near  the 
level  of  the  sea. 

To  these,  it  is  to  be  added,  that  the  western  sides  of  the  two 
great  continents  are  sensibly  warmer,  or  have  a  higher  mean 
temperature  than  the  eastern. 

(3).  The  change  in  the  density,  caused  by  change  of  tem- 
perature, produces  currents  in  the  ocean  ;  the  surface  of  water 
also  becomes  more  slowly  heated,  and  parts  with  its  heat  less 
rapidly  than  the  surface  of  land  exposed  to  an  equal  action  of  the 
sutx's  rays.  The  ocean  therefore  enjoys  a  more  equable  tempera- 
ture than  the  land,  and  influences  in  a  similar  manner  the  climate 
of  islands  and  seacoasts. 

(4),  Cultivation  appears  to  raise  the  mean  temperature,  and 
CGl*ainly  ameliorates  the  climate.  In  the  United  States,  this  effect 
appe^s  to  ke  wen  jnarke^  but  is  attended  with  an  anomaly.  The 
duratioi  Of  intense  cold  has  been  sensibly  lessened,  but  the  dimi- 
nution ot  he  iength  of  the  winter  is  wholly  in  its  earlier  part ; 
on  the  othe..  hand,  frosts  are  experienced  at  later  dates  in  the 
spring  than  foi^eriy.  §uch  are  the  more  important  circumstan- 
ces that  influence  Climate,  and  on  them  a  theory  of  the  winds  may 
be  founded. 

479.  The  air  of  our  a^osphere  receives  heat  from,  and  com- 
municates it  to,  the  parts  ^f  the  earth  on  which  it  presses.  Those 
parts  of  it  in  immediate  cou.act,  acquiring  or  parting  with  heat 
readily;  their  volumes  and  tonsions  are  therefore  changed,  a 
disturbance  of  equilibrium  takes  place,  and  motion  ensues.  Thus 
fresh  portions  of  air  are  brought  into  contact  with  the  surface  of 
the  earth,  and  the  influence  of  its  changes  of  temperature  extended. 
These  motions  in  the  atmosphere,  concur  therefore  with  those  of 
the  ocean,  of  which  we  have  already  spoken,  to  moderate  the  vi- 
cissitudes of  heat,  to  which  the  surface  would  otherwise  be  sub- 
jected. 

The  lower  stratum  of  the  air  of  the  atmosphere,  tends  in  con- 
sequence, to  an  equilibrium  of  temperature  with  the  surface  be- 
neath it ;  this  state  it  however  never  reaches,  or  never  retains 
for  more  than  a  short  space  of  time  ;  besides,  in  its  own  tendency 


Book  F/.]  THB  WINDS.  493 

to  move,  until  a  state  of  equilibrium  of  temperature  be  attained,  it 
is  set  into  a  continual,  and  frequently  violent  motion. 

This  state  of  equilibrium,  it  may  be  stated,  is  not  that  of  uni- 
form temperature  throughout ;  but  would  be  one  of  uniform  tem- 
perature at  the  mean  surface  of  the  earth,  and  of  a  temperature 
regularly  decreasing  from  that  surface  upwards,  in  conformity 
with  the  relations  of  the  air's  diminishing  density  to  specific  heat. 

Had  the  air  no  motion  growing  out  of  such  disturbances  of  tem- 
perature, its  inertia,  and  the  friction  that  takes  place  between  it 
and  the  earth,  and  among  its  own  particles,  would  cause  it  to  as- 
sume precisely  the  same  angular  velocity  with  the  part  of  the  sur- 
face immediately  beneath  it.  In  its  motions  it  must  therefore  be 
considered  as  acted  upon  by  two  forces  ;  the  one  arising  from  the 
disturbance  of  the  equilibrium  of  temperature ;  the  other,  from 
the  rotary  motion  of  the  parallel  whence  it  begins  to  move  over 
the  surface. 

From  the  foregoing  considerations  it  will  be  seen  that  the 
earth's  atmosphere  must  be  in  a  state  of  almost  constant  motion, 
forming  the  currents  that  are  styled  Winds. 

Upon  the  greater  part  of  the  surface  of  the  ocean,  these  are  re- 
ducible to  fixed  and  determinate  laws.  Upon  continents,  and  in 
high  latitudes  upon  the  ocean,  although  we  may  assign  the  gene- 
ral causes  of  the  winds,  yet  the  order  and  periods  of  their  recur- 
rences are  irregular. 

480.  The  winds  may  be  divided  into  classes,  which  we  shall 
enumerate  before  proceed  ing  to  explain  their  causes.     They  are 

1.  The  Trade  Winds; 

2.  Monsoons ; 

3.  The  local  variations  of  the  Trades  and  Monsoons  ; 

4.  The  regular  Westerly  Winds  ; 

5.  The  variable  winds  of  continents,  and  of  temperate  and  po- 
lar climates  ; 

6.  The  land  and  sea  breezes. 

The  theory  of  the  winds  has  derived  most  important  accessions 
from  the  researches  of  Daniell,  whose  labours  we  shall  make  use 
of  in  the  explanation  of  these  phenomena.  As  it  is  unnecessary 
to  enter  into  any  strict  calculations  in  relation  to  them,  we  shall, 
in  this  discussion,  dispense  with  the  use  of  algebraic  notation. 

481.  Were  the  earth  a  sphere  of  uniform  temperature,  and  at 
rest  in  space  ;  its  atmosphere  a  perfectly  dry  and  permanently  elas- 
tic fluid  ;  the  height  of  the  latter  would  be  constant  over  every  point 
of  the  earth's  surface,  and  its  density  and  elasticity,  at  equal  ele- 
vations, every  where  the  same.     The  column  of  mercury  that 
it  would  support  in  the  barometer,  would  therefore  be  the  same  at 


494  TitE  WINDS.  [Book  VI. 

every  point  on  the  surface  of  the  sphere ;  and  equal  at  equal 
heights  above  the  surface.  The  atmosphere  would  be  absolutely 
at  rest ;  and  as  its  elasticity  is  proportioned  to  the  pressure,  the 
density  would  decrease  in  geometrical  progression,  while  the  dis- 
tance from  the  surface  of  the  sphere  increases  in  arithmetical. 
When  air  is  rarefied,  its  capacity  for  heat  is  increased,  and  vice 
versa;  the  sensible  heat  of  the  atmosphere  must  therefore  decrease 
as  the  altitude  increases  ;  and  as  this  changes  the  volume  of  elastic 
fluids,  even  under  equal  pressures,  the  barometer  alone  will  no 
longer  be  the  exact  measure  of  the  progressive  density,  but  must 
be  associated  with  the  thermometer.  Any  change  of  temperature 
that  affects  every  part  of  the  sphere,  would  cause  an  increase  in 
the  elasticity  of  the  atmosphere,  and  in  its  consequent  height, 
without  producing  any  motion  in  the  lateral  direction,  or  any 
change  in  the  pressure  upon  the  surface;  but  the  pressure  will  be 
changed  at  all  other  altitudes. 

If  the  temperature  of  the  sphere,  instead  of  being  equal  at  every 
point,  were  greatest  at  the  equator,  and  decreased  towards  the 
poles,  the  pressure  on  every  point  of  the  surface  would  still  con- 
tinue the  same  ;  but  the  altitude  of  the  atmospheric  column  would 
become  greatest  at  the  equator,  and  its  specific  gravity  at  the  sur- 
face less  there  than  at  the  poles.  The  heavier  fluid  at  the  poles 
must,  by  its  greater  weight  pass  beneath,  and  displace  the  lighter, 
and  a  current  will  be  established  in  the  lower  part  of  the  atmos- 
phere, from  the  poles  towards  the  equator.  The  difference  in  the 
specific  gravity  of  the  polar  and  equatorial  columns  becomes  less 
as  \ve  ascend  into  the  atmosphere  ;  while  the  elasticity,  which  is 
constant  at  the  surface,  varies  with  the  height,  and  the  barometer 
stands  higher  at  equal  elevations  in  the  equatorial,  than  in  the 
polar  column.  It  will  hence  happen,  that,  at  some  definite  height, 
the  unequal  density  of  the  lower  strata  will  be  compensated  ;  and 
a  counter-current  will  take  place  in  the  higher  regions  from  the 
equator  towards  the  poles. 

The  heights  at  which  this  would  happen,  under  certain  circum- 
stances may  be  calculated,  and  the  velocity  of  each  current  deter- 
mined. This  has  been  done  by  Daniell,  to  whose  work  the  reader 
is  referred,  for  the  process  and  inferences.  From  his  investi- 
gations it  appears,  that  the  lower  current  directed  from  the  poles 
towards  the  equator,  extends  to  the  height  of  two  miles  and  a  half, 
gradually  diminishing  in  velocity  from  the  surface  upwards.  At 
the  last  mentioned  height  the  counter  current  begins,  and  its  ve- 
locities gradually  increase  from  that  altitude  upwards. 

The  velocity  and  direction  of  these  currents  may  be  affected  by 
the  partial  rarefaction  or  condensation  of  any  of  the  columns  ;  and 
such  change  of  density  will  naturally  take  place,  in  consequence 


Book  VL]  TUB  WINDS.  495 

of  the  vicissitudes  of  the  seasons,  and  the  alternations  of  day  and 
night. 

If  the  sphere  revolve  around  its  polar  diameter,  as  an  axis,  an 
apparent  modification  will  take  place  in  the  direction  of  the  cur- 
rents. The  lower  current,  coming  from  a  point  whose  velocity 
of  rotation  is  less  than  that  at  which  it  arrives,  will  appear  to  be 
affected  with  a  motion,  in  a  direction  contrary  to  that  of  the  revo- 
lution of  the  sphere  ;  while  the  upper  current,  being  under  op- 
posite circumstances,  will  be  apparently  affected  in  an  opposite 
manner. 

The  earth  revolves  around  its  axis  in  a  direction  from  west  to 
east ;  and  hence  the  great  equatorial  currents,  that  are  in  fact  di- 
rected, on  the  north  side  of  the  equator,  from  north  to  south,  and 
on  the  south  side  from  south  to  north,  appear  in  both  cases  de- 
fleeted  towards  the  east. 

In  the  months  of  April  and  October,  such  a  state  of  things  does 
actually  take  place  upon  the  earth  ;  hence  N.  E.  winds  prevail 
at  those  periods  throughout  the  whole  northern,  and  S.  E.  winds 
throughout  the  whole  southern  hemisphere  ;  the  hemispheres 
being  divided  by  the  equator  of  temperature,  and  not  by  that  of 
latitude. 

At  other  seasons,  the  regular  law  of  decreasing  temperature  is 
interrupted  even  upon  the  surface  of  the  ocean,  at  latitudes  of  from 
28°  to  32° ;  and  is  not  to  be  recognised  upon  the  land  ;  hence 
these  winds  are  constant  only  within  these  limits,  and  in  the 
open  ocean.  These  constant  currents  are  called  the  Trade  Winds, 
and  from  their  directions,  the  N.  E.  and  S.  E.  Trades. 

The  velocity  of  rotation  changes  more  for  a  given  difference  of 
latitude  in  high  than  in  low  latitudes;  hence  the  apparent  devia- 
tion, from  a  true  northern  or  southern  direction,  will  be  greatest 
near  the  outer  verge  of  the  trade  winds,  and  least  near  their  cen- 
tral zone. 

4S2.  This  central  zone  that  divides  the  trade  winds,  has  a 
breadth,  varying  at  different  seasons,  from  2\  to  9  degrees.  It 
corresponds  with  the  equator  of  temperature,  and  hence  varies 
in  position,  §  474,  a  few  degrees,  but  is  always  on  the  northern 
side  of  the  equator.  Within  this  narrow  zone  the  winds  are 
subject  to  no  regular  law,  and  hence  it  is  said  to  be  the  seat  of 
the  Variables.  In  this  space  the  velocities  of  the  currents  pro- 
ceeding in  opposite  directions  destroy  each  other,  and  an  accu- 
mulation, as  has  been  stated,  would  take  place,  did  not  the  air 
rise  and  join  the  counter-current,  that  continually  flows  in  the 
higher  regions. 

At  the  outer  limits  of  the  regular  trades,  it  might  be  inferred 
that  the  descent  of  the  counter-current  would  form  a  narrow  zone. 


496  THE  WINDS.  [Book  VI. 

of  winds  uncertain  in  direction,  and  generally  light ;  such  a  zone 
is  distinctly  marked,  and  well  known  in  the  North  Atlantic 
Ocean.  Whether  it  be  found  in  the  Pacific  and  South  Atlantic 
Oceans  cannot  be  stated,  for  the  want  of  careful  and  sufficient 
recorded  observations. 

The  trade  winds,  as  may  be  inferred  from  this  theory,  prevail 
in  the  open  ocean,  in  the  Atlantic  and  Pacific,  between  the  lati- 
tudes of  30°  N.  and  27°  S.  On  entering  them  from  either  side, 
the  deviation,  growing  out  of  the  rotation  of  the  earth,  towards 
the  east,  is  greatest ;  and  this  deviation  becomes  less  and  less  as 
the  equatorial  zone  is  approached  ;  in  the  immediate  vicinity  of 
this  zone,  the  wind  is  nearly  due  N.  on  the  northern  side  of  the 
equator,  and  nearly  due  S.  on  the  southern. 

483.  In  the  Indian  Ocean,  winds  changing  their  direction  half 
yearly,  and  blowing  regularly  in  each  direction  for  nearly  six 
months,  are  experienced.  These  periodic  winds  are  called  the 
Monsoons.  The  cause  of  them  is  to  be  found  in  the  position  of 
this  ocean  in  respect  to  the  adjacent  continents. 

To  the  north  of  the  Indian  Ocean  extends  the  whole  mass  of 
the  old  continent,  with  the  exception  of  the  southern  extremity 
of  Africa.  The  ocean  and  the  land,  thus  placed,  are  acted  upon 
by  the  sun,  in  his  annual  course,  with  different  degrees  of  inten- 
sity at  different  seasons.  In  the  summer  of  northern  latitudes, 
the  sun  is  vertical  over  large  portions  of  the  continent,  and  ac- 
cording to  the  principles  of  §  476,  the  superficial  heat  of  the 
land  being  more  speedily  raised,  even  by  an  equal  exposure  to 
the  sun,  becomes  greater  than  that  of  the  ocean.  The  denser  air 
at  the  surface  of  the  ocean  therefore  presses  towards  the  land, 
causing  a  current  whose  absolute  motion  is  from  south  to  north. 
On  the  northern  side  of  the  equator,  coming  from  a  point  whose 
velocity  of  rotation  is  greater  than  that  of  the  points  it  meets  in 
its  course,  it  has  an  apparent  deflection  towards  the  west  and 
forms  a  S.  W.  wind.  Hence  in  the  Indian  Ocean,  the  south- 
western monsoon  blows  between  the  months  of  April  and  No- 
vember, on  the  north  side  of  the  equator. 

The  causes  that  produce  the  S.  W.  monsoon,  also  operate  on 
the  southern  side  of  the  equator,  as  far  as  11°  S.  The  current 
they  cause,  pressing  N.  towards  the  Equator,  from  a  parallel 
that  has  a  less  velocity  of  rotation,  appears  as  a  S.  E.  wind. 

When  the  sun  is  on  the  southern  side  of  the  equator,  the  old 
continent,  losing  by  radiation  more  heat  than  it  receives,  becomes 
colder  than  the  Indian  Ocean.  The  air  above  it,  therefore, 
presses  to  the  south,  and  the  influence  extends  as  far  as  11°  S. 
For  reasons  the  converse  of  the  preceding,  the  apparent  direction 
becomes  N.  E.  on  the  northern  side  of  the  equator,  and  N.  W 


Book  PL]  THE  WINDS.  4i>7 

on  the  southern.  To  the  south  of  this  parallel  the  regular  S.  E. 
trade  wind  blows  continually,  in  the  Indian  as  well  as  in  the  other 
Oceans. 

484.  The  most  important  modifications  of  the  trade  winds, 
growing  out  of  local  circumstances,  are  as  follows  : 

The  continent  of  Africa,  over  which  the  sun  is  continually 
vertical,  is  always  heated  at  the  surface,  for  reasons  already  as- 
signed, §  476,  to  a  temperature  higher  than  the  adjacent  ocean. 
Hence,  in  the  Gulf  of  Guinea,  a  wind  sets  almost  constantly  to- 
wards the  land,  and  is  modified  in  its  direction  by  the  tending  of 
the  shore.  Between  the  region  in  which  this  sea  breeze  blows, 
and  that  in  which  the  trade  winds  begin  again  to  prevail,  these 
two  winds,  diverging  from  the  same  space,  cause  an  exhaustion, 
which  is  supplied  by  a  counter-current  in  higher  parts  of  the  at- 
mosphere. Within  this  interval  there  is  a  portion  of  the  surface 
of  the  ocean  that  is  the  seat  of  almost  perpetual  calms. 

The  course  of  the  trade  winds  is  interrupted  by  the  continent 
of  America,  hence  their  influence  is  not  felt  until  at  some  dis- 
tance from  the  coast  of  the  Pacific  Ocean. 

Upon  the  eastern  coast  of  North  America,  in  the  summer  sea- 
son of  the  respective  hemispheres,  the  greater  heat  of  the  land 
draws  a  current  from  the  ocean ;  by  this  the  extent  of  the  trade 
wind  is  increased.  The  course  of  this  part  is,  however,  E.  or 
even  S.  E.  on  the  coast  of  Florida,  Georgia,  &c. 

The  monsoons  in  the  neighbourhood  of  the  land,  have  their 
courses  deflected  also,  and  sometimes  their  influence  merges  alto- 
gether in  the  land  and  sea  breeze. 

485.  Between  the  parallels  of  30°  and  40°,  in  both  the  North 
and  South  Pacifies,  a  westerly  wind  blows  almost  constantly  ; 
intermitting  only  for  a  short  space  of  time,  after  each  equinox, 
when  a  regular  distribution  of  temperature,  over  the  whole  earth, 
gives  rise  to  the  N.  E.  and  S.  E.  trade  winds. 

In  the  Northern  Atlantic  this  wind  is  not  constant,  in  conse- 
quence of  the  comparative  want  of  breadth  of  that  ocean,  by 
which  it  is  subjected  to  the  influence  of  the  contiguous  continents. 
A  westerly  wind  is,  however,  the  prevailing  wind  in  this  ocean, 
except  in  the  months  of  April  and  October,  when  a  N.  E.  wind 
is  more  frequent. 

The  cause  of  the  existence  of  such  westerly  currents  may  be 
thus  explained. 

Within  the  tropics,  and  to  a  short  distance  beyond  them,  the 
variations  of  temperature,  from  a  law  of  regular  decrease  from 
the  equator  towards  the  poles,  are  so  small  as  to  be  insensible, 
and  hence,  as  has  been  stated,  the  trade  winds  are  constant  within 

63 


498  THE  WINDS.  [Book  VI. 

certain  limits.  Without  these  limits,  the  parallels  are  alternately 
warmer  and  colder,  according  to  the  reason,  than  would  be  con- 
sistent with  the  law  of  regular  decrease.  In  some  one  parallel, 
the  deviation  from  this  law  will  be  the  greatest.  This  may  be 
taken  as  about  the  parallel  of  40°,  in  which,  as  may  be  seen  from 
the  examples  of  New-York  and  Pekin,  the  vicissitudes  of  tem- 
perature are  excessive.  When  this  parallel  is  more  heated  than 
is  consistent  with  the  law  of  the  mean  temperatures,  the  course 
of  the  great  current  from  the  poles  towards  the  equator  is  inter- 
rupted, the  atmosphere  in  contact  with  the  surface  of  the  earth 
will  be  accelerated  on  the  side  of  this  parallel  nearest  to  the  pole  ; 
on  the  side  nearest  the  equator,  the  air  increases  in  density,  and 
hence  moves  in  a  direction  contrary  to  that  which  it  would  have 
if  the  temperature  decreased  regularly  towards  the  poles.  To 
counteract  the  condensation  that  would  hence  arise  in  the  paral- 
lel of  40°,  a  counter-current  takes  place.  The  lower  current 
coming  from  a  parallel  whose  velocity  of  rotation  is  greater  than 
that  which  it  reaches,  is  apparently  impressed  with  a  motion 
from  W.  to  E.  and  becomes  a  S.  VV.  wind  on  the  north  side  of 
the  equator,  and  N.  W.  on  its  southern  side. 

When  this  parallel  has  a  lower  temperature  than  is  consistent 
with  the  law  of  regular  decrease,  it  has  been  demonstrated  by 
Daniell,  that  the  atmospheric  pressure  would  be  diminished  ; 
for  this  reason  a  current  would  set  toward  it  on  both  sides,  in  or- 
der to  restore  the  equilibrium,  and  thus  the  two  causes  so  different 
in  themselves,  will  produce  similar  effects,  and  winds  deflected 
towards  the  west  will  again  take  place. 

The  slow  changes  that  take  place  in  the  temperature  of  the  sur- 
face of  the  ocean,  growing  out  of  the  causes  stated  in  §  476,  make 
the  parallel  in  which  these  opposite  influences  operate  to  produce 
this  effect,  nearly  constant  in  position.  And  it  is  for  a  similar 
reason,  that  the  monsoons  do  not  vary  gradually  in  intensity  and 
direction  with  the  declination  of  the  sun,  but  intermit  wholly 
for  a  time,  and  then  assume  the  new  direction. 

The  interval  of  the  monsoons  is  attended  with  great  oscillations 
in  the  atmosphere;  great  accumulations  tnke  place  in  some  places, 
attended  with  corresponding  rarefactions  in  others  ;  these  mu- 
tually re-act  upon  each  other  ;  thus  violent  storms,  the  Typhoons 
of  the  Indian  seas,  occur  in  the  interval  of  the  monsoons. 

486.  Upon  the  continents, thechangesoftemperature  from  day 
to  day,  and  the  alternations  of  heat  from  day  to  night,  are  rapid  and 
frequent;  hence  there  is  no  constancy  in  the  direction  or  inten- 
sity of  the  winds.  In  high  latitudes,  even  in  the  open  sea,  simi- 
lar inequalities  occur.  Hence,  the  land  and  ocean,  in  lati- 
tudes higher  than  40°,  are  the  seat  of  winds  that  can  be  reduced 


Rook  VL~\  THE  WINDS.  499 

to  no  fixed  laws,  and  the  frequency  of  whose  changes  increases 
with  the  latitude. 

In  the  blowing  of  these  variable  winds,  the  inertia  of  the  air 
tends  to  cause  accumulations  in  the  parts  towards  which  they 
blow,  and  expansions  in  those  whence  they  come  ;  the  elastic 
nature  of  the  air  allows  these  to  increase,  until  the  moving  force 
is  destroyed,  when  a  returning  current  is  formed  which  will  again 
cause  similar  condensations  and  exhaustions.  Thus  the  varia- 
tions in  the  height  of  the  barometer,  which  have  been  noted  in 
§  385,  become  of  greater  extent  in  high,  than  in  low  latitudes; 
and  when  winds  have  gradually  expended  their  force,  a  wind  in 
a  direction  exactly  contrary  often  succeeds. 

Although  the  variable  winds  of  temperate  climates  are  subject 
to  no  fixed  laws,  still  we  may  often  find  in  the  local  circum- 
stances of  countries,  reasons  why  certain  winds  should  blow  more 
frequently  than  others.  Such  winds  are  called  the  Prevailing 
Winds,  of  the  particular  climate. 

In  the  northern  and  middle  portions  of  the  seaboard  of  the  Uni- 
ted States,  the  great  prevailing  winds  are  the  N.  W.,  the  N.  E  ,the 
S.  W.,  and  the  S.  E.  By  an  attentive  examination  of  the  cir- 
cumstances of  the  country,  we  may  easily  show  why  these  should 
be  of  frequent  occurrence,  and  probably  prevail  to  the  exclusion  of 
all  others. 

A  great  current  of  the  ocean  called  the  Gulf  Stream,  proceeds 
from  the  Gulf  of  Mexico,  and  runs  nearly  parallel  to  the  coast  of 
North  America,  as  far  as  the  banks  of  Newfoundland.  This 
current  during  the  winter  months,  is  much  warmer  than  the 
neighbouring  continent ;  hence  a  current  of  air  frequently  sets 
from  the  land  towards  the  ocean,  which  forms  the  N.  W.  wind 
of  the  United  States. 

In  summer,  although  the  land  becomes  warmer  than  the  Gulf 
Stream,  the  great  difference  of  temperature  between  the  sea- 
board and  the  interior,  in  which  at.  the  lat.  of  60°,  and  at  the 
depth  of  six  feet  beneath  the  surface,  the  ground  is  entirely  frozen, 
is  sufficient  to  account  for  the  N.  W.  being  a  frequent  wind. 

The  trade  winds  are  interrupted  in  their  course  by  the  great 
chain  of  mountains  that  traverse  nearly  the  whole  continent  of 
America.  This  interruption  causes  an  accumulation  of  air 
against  their  sides.  It  cannot  be  lessened  by  a  return  in  the  di- 
rection whence  it  came,  in  consequence  of  the  perpetual  current 
of  the  trade  winds.  It  will,  therefore,  when  no  other  cause  for 
a  prevailing  wind  exists,  press  over  the  whole  continent;  and,  if 
the  accumulation  have  been  going  on  fora  long  time,  will  exert  a 
force  that  no  other  wind  of  our  climate  does. 

TheN.  E.  wind  of  the  United  States,  as  well  as  that  of  Europe, 


500  THE  WINDS.  [Book    VI. 

maybe  considered  as  the  great  current,  directed  towards  the  equa- 
tor, and  exerting  its  influence  when  no  other  cause  is  in  action. 
That  it  should  be  frequent,  may  be  explained  from  the  fact,  that 
the  cleared  and  cultivated  portions  of  the  United  States,  will  often 
be  under  the  circumstances  of  a  regular  increase  of  temperature, 
from  the  N.  E.  to  the  S.  W.  This  wind  was  formerly  confined 
to  a  strip  extending  but  a  few  miles  from  the  coast.  As  the  coun- 
try has  been  cleared,  its  influence  has  been  more  extended,  and  it 
is  annually  becoming  more  prevalent. 

If  a  N.  W.  wind  have  prevailed  for  some  time,  as  a  current  of 
air  must,  according  to  the  law  of  lateral  communication  of  mo- 
tion, follow  the  direction  of  the  Gulf  Stream,  an  accumulation 
must  take  place  over  the  latter,  which,  seeking  to  restore  the  equi- 
librium of  pressure,  presses  towards  the  continent,  and  causes  a 
S.  E.  Wind. 

487.  The  land  and  sea  breezes  are  winds  whose  period  is  a 
single  day,  the  former  prevailing  when  the  sun  is  below,  the  latter 
when  he  is  above  the  horizon.  The  causes  are  to  be  sought  in  the 
different  manner,  in  which  land  and  water  are  affected  by  radia- 
tion, and  the  direct  heat  of  the  sun.  During  the  day,  the  surface 
of  the  land,  as  will  be  seen  by  reference  to  §  170,  becomes  more 
heated  than  that  of  the  adjacent  ocean.  Hence  a  current  begin- 
ning at  some  hour  in  thannorning,  and  continuing  until  the  sun 
is  near  setting,  will  flow  from  the  water  towards  the  land.  At 
night  the  water  remains  warm,  while  the  surface  of  the  land  cools 
rapidly,  and  hence  the  current  sets  from  the  land  towards  the 
water. 

Of  all  the  winds  in  the  climate  of  New- York,  a  north  wind  is 
perhaps  the  most  rare.  It  however  sometimes  blows  for  two  or 
three  days  together,  and  is  remarkable  for  the  extreme  heat  with 
which  it  is  attended.  The  cause  of  this  high  temperature  seems 
to  be,  that  a  north  wind  prevents  the  action  of  the  sea  breeze,  that 
would  otherwise  act  to  temper  the  climate,  and  moderate  its  vicis- 
situdes. 

The  variable  winds  of  temperate  climates,  as  maybe  seen  from 
what  has  been  said,  arises  either  from  a  condensation  in  some  part 
of  the  atmosphere,  or  a  rarefaction  in  others,  or  both  may  concur. 

When  the  former  is  the  cause,  the  wind  proceeds  forward,  ex- 
tending itself  in  the  direction  towards  which  it  appears  to  blow. 
Such  is  the  case  with  the  N.  W.  wind  of  our  climate. 

Within  the  tropics,  the  islands  and  countries  situated  upon 
the  sea-coast,  have  their  climates  tempered  by  winds  styled  the 
Land  and  Sea  Breezes.  The  sea  breeze  begins  to  blow  three  or 
four  hours  after  sunrise,  and  continues  to  blow  until  a  little  after 
sunset.  The  land  breeze  commences  three  or  four  hours  after 


Book   VI.}  THE  WINDS.  501 

sunset,  and  continues  until  about  sunrise.  The  cause  of  these 
alternating  winds  may  be  thus  explained  :  The  land  and  sea, 
being  both  equally  exposed  to  the  action  of  the  sun  during  the 
day,  the  former  as  explained  in  §476,  becomes  more  heated  at 
the  surface  than  the  latter.  The  air  in  contact  with  the  land,  in 
consequence  expands  and  rises,  while  that  over  the  sea  presses  in 
to  supply  the  place.  At  night,  the  surface  of  the  land  parts  with 
its  heat  most  rapidly,  and  the  course  of  the  current  is  reversed. 

These  winds  are  not  confined  to  the  tropics,  but  may  be  ob- 
served on  the  sea-coast  of  countries  situated  in  latitudes  as  high 
as  45°,  during  the  summer  of  the  hemisphere  in  which  they  are 
situated.  Thus,  at  New- York,  a  sea  breeze  is  experienced  almost 
daily  during  the  months  of  June,  July,  and  August ;  and  a  land 
breeze  may  be  occasionally  observed  during  the  same  months. 

When  the  latter  cause  occurs,  the  wind  will  appear  to  blow  first 
in  the  quarter  towards  which  it  is  directed.  Such  is  the  N.  E. 
wind  of  our  climate,  which  begins  to  blow  in  Florida  many  hours 
before  it  is  felt  at  Boston. 

The  manner  in  which  a  wind  of  this  character  may  arise,  and 
thus  extend  itself  to  witidward,  may  be  illustrated  by  referring  to 
what  occurs  on  opening  the  gate  of  a  sluice,  in  which  case  the 
current  sets  towards  the  opening  ;  but  the  motion  begins  in  im- 
mediate contact  with  the  sluice,  and  is  propagated  in  a  direction 
contrary  to  the  current. 


502  MOTION  OF  VAPOUR  IN  [Book    VI. 

CHAPTER  XIV. 

OF  THE  MOTION  OP  VAPOUR  IN  THE  ATMOSPHERE. 

488.  It  will  be  seen  by  reference  to  §  370,  that  water  forms 
vapour  at  all  temperatures  whatsoever,  of  a  tension  and  density 
having  relation  to  the  temperature,  according  to  the  tables  of  §  374 
and  §  379.     Now,  as  a  great  portion  of  the  surface  of  the  earth  is 
covered  with  that  liquid,  it  follows,  that  vapour  will  rise  from 
it,  and  by  the  general  property  of  elastic  fluids  to  form  atmos- 
pheres independent  of  each  other,  will  tend  to  distribute  itself 
over  the  surface  of  the  earth  ;  and  it  would  assume  the  tension  and 
elasticity  due  to  the  temperature  of  the  space  it  occupies,  did  no 
opposing  force  act  to  prevent  it. 

489.  Were  the  earth  a  sphere,  wholly  covered  with  water,  of 
uniform  temperature  throughout  its  surface,  and  if  we  suppose 
the  aerial  atmosphere  not  to  exist,  the  water  would  form  an  atmos- 
phere of  vapour,  whose  pressure  would  be  equal  to  the  elastic 
force  of  vapour  at  the  constant  temperature  of  the  surface.     The 
temperature  of  this  aqueous  atmosphere  would  not  be  uniform 
throughout,  but  would  be  so  only  at  the  surface,  for  the  higher  por- 
tions undergoing  less  pressure  would  expand,  and  their  tempera- 
ture would  diminish  to  that  corresponding  to  the  tension  and  den- 
sity of  the  vapour  at  the  given  point.  To  take  an  instance  :  Were 
a  sphere  whose  surface  is  wholly  covered  with  water,  and  which 
has  no  aerial  atmosphere,  to  have  at  every  point  on  its  surface  the 
temperature  of  32°  ;  an  atmosphere  of  vapour  would  be   formed 
around  it,  whose  tension  would  be  0.2  inches,  and  whose  tem- 
perature at  the  surface  would  be  32°.     At  the  altitude  of  30,000 
feet  it  may  be  calculated  that  the  tension  would   be  diminished 
one-half,  or  to  0. 1  inch,  and  the  temperature  of  the  vapour  to 
13°. 

The  atmosphere  of  vapour  would  be  in  perfect  equilibrium,  and 
at  rest,  over  the  whole  surface  of  the  sphere  ;  and  would,  by  its 
pressure,  prevent  the  formation  of  any  more  vapour.  No  preci- 
pitation would  occur  in  any  part,  and  the  whole  mass  would  be 
clear  and  transparent. 

An  uniform  increase  of  the  temperature  of  the  surface,  would 
cause  the  formation  of  new  vapour ;  the  tension  of  the  whole 
would  become  uniform  ;  the  temperature  of  its  lower  parts  would 
be  the  same  as  that  of  the  surface  of  the  sphere,  and  an  analogous 
decrease  of  temperature  and  tension  would  take  place  at  increas- 
ing elevations. 


»--;"  ***'"'  *: 

; 

Book    K/.]  ,THE   ATMOSPHERE.  503 

"  i-' 

490.  But  if  the  temperature  of  the  sphere  were  to  become 
unequal,  the  circumstances  would  be  different.     If  we  assume  it 
to  follow  the  law  of  its  mean  temperature,  being  warmest  at  the 
equator,  and  to  decrease  in  heat  according  to  some  definite  law, 
from  the  equator  to  the  poles ;  the  tension  of  the  vapour  over  the 
whole  surface,  would  be  due  to  the  minimum  temperature,  §372, 
or  to  that  of  the  poles;   but  the.  evaporation  being  due  to  the  heat 
of  the  different  points  on  the  surface,  would  be  determined  by 
that  heat,  and  go  on  continually  ;  while  at  the  poles,  an  equal  and 
rapid  condensation  would  take  place.     The  vapour  would  in  this 
case,  flow  in  mass,  from  the  equator  towards  the  poles,  and  the 
precipitation  at  the  latter  points  would  raise  the  level  of  the  ocean, 
until  currents  were  formed,  by  which  all  the  condensed  water 
would  flow  back  to  the  equator. 

491.  If  some  retarding  force  were  to  act,  by  which  the  flow  of 
vapour  is  resisted  in   its  course  from  the  equator   towards  the 
poles  ;    the  precipitation  would  be   distributed  throughout  the 
whole  sphere,  except  at  the  zone  of  greatest  temperature.     Con- 
tinual evaporation  would  go  on  at   the  equator;  continual  pre- 
cipitation at  the  poles,  and  both  evaporation  and  precipitation  at 
all  intermediate  latitudes. 

Such  a  retarding  force  is  to  be  found  in  the  aerial  atmosphere. 
The  vapour,  although  it  constantly  tends  to  form  an  atmosphere, 
according  to  its  own  mechanical  laws,  is  resisted  in  its  motions 
by  the  aerial  atmosphere  through  which  it  is  compelled  as  it 
were  to  filter ;  and  thus,  were  the  circumstances  of  which  we 
have  already  spoken,  to  be  all  that  affect  the  mixed  mass  of  air 
and  vapour,  condensation  would  be  taking  place  in  the  higher  re- 
gions at  all  latitudes,  attended  at  the  same  time  with  evaporation 
from  beneath. 

492.  The  relations  of  air  under  different  pressures  to  heat,  are 
different  from  that  of  vapour,  and  the  temperature  of  an  aerial  at- 
mosphere diminishes  much  more  rapidly  with  increasing  eleva- 
tions,  than  that   of  an  aqueous  atmosphere;  here  then,  we  find 
upon  the  principles  of  §368,  a  new  cause  of  precipitation,  which 
would  in  the  higher  regions  be  attended  with  a  corresponding  in- 
crease in  the  evaporation  from  beneath. 

If  then  the  earth  were  wholly  covered  with  water,  continual 
rains,  or  at  least  perpetual  clouds,  would  be  experienced  every 
where  but  at  the  equator. 

493.  As,  however,  rather  more  than  one-fourth  part  of  the 
earth's  surface  is  dry  land  ;  this  produces  a  very  marked  change 
in  the  circumstances  of  the  atmosphere.     The  land  furnishes  a 
comparatively  small  quantity  of  vapour.    Hence,  as  vapour  tends 


504  MOTION  OF  VAPOUR  IN  [Book  VI. 

\ 

to  fojrra  an  atmosphere  of  itself,  distributed  according  to  the  rela- 
tions of  temperature  and  tension,  over  the  whole  surface,  the  va- 
pour formed  over  the  surface  of  the  ocean  would  continually  press 
towards  the  land,  until  a  state  of  equilibrium  could  be  attained. 

If  the  land  be  warmer  than  the  ocean,  the  vapour  would  be 
heated  above  its  original  temperature,  and  a  greater  quantity  by 
weight,  could  exist  without  deposition  in  a  gvven  space;  hence, 
the  vapour  that  might  otherwise  be  precipitated  over  the  ocean, 
'  would  be  diverted  towards  the  land,  and  even  there  no  deposit 
might  ensue. 

If  the  land  be  colder  than  the  adjacent  ocean,  vapour  will  still 
flow  towards  it,  but  it  will  now  be  condensed  upon  it,  and  a  part 
at  least  of  the  condensation  that  would  otherwise  take  place  upon 
the  ocean,  will  take  place  upon  the  land. 

The  ocean  has  a  temperature  far  less  variable  than  that  of  the 
land,  and  thus  both  will  be  affected  with  alternations  of  rain  and 
sunshine,  according  to  the  relations  between  their  temperatures. 
In  the  temperate  and  frigid  zones,  these  will  be  subject  to  no 
fixed  laws  ;  but  in  the  torrid  zone,  seasons  of  considerable  length 
will  be  wet  or  dry,  according  to  the  latitude  of  the  place  and  the 
declination  of  the  sun. 

494.  The  flow  of  the  vapour,  in  conformity  with  its  own  me- 
chanical laws,  is  not  only  retarded  by  the  mere  resistance  of  the 
atmosphere,  but  is  affected  by  the  winds.     When  the  course  of 
the  wind  coincides  with  the  direction  the  vapour  would  assume 
under  its  own  pressure,  the  flow   of  the  vapour  is  accelerated  ; 
when  the.  contrary  is  the  case,  it  is  retarded  ;  and   it  may  thus 
happen,  that  some  districts  of  the  continents  are  wholly  deprived 
of  moisture,  while  others  receive  a  superabundant  proportion. 

Winds  also  agitate  and  mix  together  masses  of  air  of  different 
temperatures,  and  when  these  contain  moisture,  of  a  tension  ap- 
proaching to  the  maximum  due  to  their  respective  temperatures, 
precipitation  must  almost  always  ensue.  That  this  must  be  the 
case,  will  appear  from  the  consideration,  that  the  quantity  of  va- 
pour that  can  exist  in  a  given  space,  varies  in  geometric  progres- 
sion, while  the  temperature  varies  in  arithmetic  ;  and  the  tempera- 
ture that  results  from  the  mixture  of  equal  masses  of  air,  is  the 
arithmetic  mean  of  their  respective  temperatures.  As  the  latter 
is  always  greater  than  the  geometric  mean,  the  quantity  of  vapour, 
if  both  masses  of  air  approached  to  saturation,  will  be  greater  than 
is  consistent  with  the  resulting  temperature,  and  the  excess  must 
be  precipitated. 

495.  It  will  be  obvious,  from  what  has  been  stated,  that  vapour 
is  in  almost  all  cases  pressing  from  the  ocean  towards  the  land  ; 


Book   F7.J  -THE  ATMOSPHERS.  «f-,*  405 

while  upon  the  latter  a  precipitation  must  ensue,  often  greats* 
than  upon  an  equal  surface  of  the  ocean.  In  this  excess  of  pre- 
cipitation we  are  to  seek  the  origin  of  springs  and  rivers ;  by  ihe 
latter  this  excess  is  restored  to  the  sea,  to  be  again  evaporated, 
and  thus  keep  up  the  continual  circulation. 

496.  So  long  as  aqueous  matter  remains  in  the  state  of  vapour, 
it  is  transparent.  On  its  first  condensation  a  cloud  appears.  The 
manner  of  the  formation  of  .clouds  is  as  follows  :  Water  on  its 
first  condensation  tends  to  unite  in  the  form  of  hollow  globules, 
or  vesicles  containing  air;  as  it  parts  at  the  same  time  with  its 
latent  heat,  the  air,  as  well  within  the  vesicles  as  between  them, 
is  rarefied,  and  the  united  mass  of  water  and  rarefied  air  may  re- 
main as  light  as  an  equal  bulk  of  atmospheric  air,  or  even  lighter. 
Clouds  may  therefore  remain  floating  in  the  atmosphere,  or  even 
rise.  As  this  heat  is  dissipated,  the  clouds  grow  heavier  and  fall, 
while  the  air  in  the  vesicles  losing  its  elasticity,  permits  them 
to  be  broken  by  the  internal  pressure.  The  water  then  runs  into 
drops,  which,  being  many  times  heavier  than  atmospheric  air, 
descend,  forming  Rain. 

Clouds  may  be  formed  in  all  cases  where  the  temperature  of 
the  ground  is  lower  than  that  at  which  the  vapour,  mixed  with 
atmospheric  air,  can  remain  permanent.  Thus:  whenever  a 
warm  wind  flows  over  a  cold  surface,  mists  and  fogs  take  place  ; 
and  if  the  difference  of  temperature  be  considerable,  they  may 
break  into  rain.  For  an  equal  difference  of  temperature  between 
the  ground  and  air,  it  may  be  shown  by  calculations  formed  on 
the  table  of  §  308,  that  the  greatest  quantity  of  precipitation  will 
take  place,  when  the  two  unequal  temperatures  are  both  high. 
Thus  the  causes  that  would  produce  heavy  rains  in  warm  climates, 
may  produce  no  more  than  fogs,  or  dense  mists,  in  those  that  are 
colder. 

Clouds  may  also  be  formed  on  sudden  changes  of  wind,  upon 
the  principle  explained  in  §491,  when  two  masses  of  air  are 
mixed  that  are  both  nearly  saturated  with  moisture.  It  is  to  this 
cause  that  nearly  all  the  rains  of  temperate  climates  are  due.  The 
passing  of  warm  winds  over  cold  surfaces,  rarely  produces  more 
than  mists  or  fogs,  except  in  warm  climates. 

When  clouds,  after  being  formed,  begin  to  descend,  in  conse- 
quence of  the  dissipation  of  the  heat,  by  the  rarefaction  arising 
from  which  they  are  supported,  they  often  reach  strata  of  the 
atmosphere  comparatively  dry,  and  of  higher  temperature  than 
they  themselves  possess.  In  such  a  case,  the  vapour  may  be  again 
taken  up,  and  the  cloud  dissipated.  Thus  clouds  are  frequently 
seen  to  roll  down  the  sides  of  the  mountains,  and  to  disappear  at 
a  certain  level ;  this  is  a  proof  of  a  dry  state  of  the  air  beneath, 

64 


506  MOTION  OP  VAPOUR  IN  [Book  VI. 

and  is  therefore  considered  by  the  inhabitants  of  mountainous 
eountries,  as  a  prognostic  of  fair  weather. 

When  a  cloud,  on  the  other  hand,  formed  in  high  and  cold  re- 
gions, passes  in  its  descent  through  strata  saturated  with  moisture, 
or  nearly  so,  it  may  cool  them  until  precipitation  ensue;  the  pre- 
cipitated moisture,  uniting  itself  to  the  descending  cloud,  will 
augment  the  intensity  of  the  rain  it  causes.  Thus  the  same  rain 
will  be  more  copious  in  vallies,  than  upon  the  neighbouring  moun- 
tains ;  and  the  difference  is  so  sensible^n  this  respect,  that  it  has 
been  detected  by  means  of  the  rain-gauges  at  the  observatory  of 
Paris,  one  of  which  is  upon  the  ground,  the  other  upon  the  ter- 
raced roof  of  the  building. 

497.  When  the  precipitation  of  vapour  ensues  at  temperatures 
below  the  freezing  point,  Snow  is  formed  ;  the  particles  of  the  con- 
densed aqueous  matter  being  free  to  move  in  any  direction,  ar- 
range themselves  under  the  action  of  their  mutual  attraction,  in 
the  manner  of  crystals.     These  crystals  have  usually  the  figure  of 
six-pointed  stars  ;  and  the  aggregation  of  broken  crystals  of  this 
shape  forms  flakes  of  snow. 

498.  Hail  is  a  phenomenon  that  is  not  completely  explained; 
the  best  theory  on  the  subject,  although  not  absolutely  satisfactory, 
is  as  follows  : 

It  is  known  that  when  water  is  frozen  in  a  torricellian  vacuum, 
it  granulates  and  assumes  the  form  of  hail  ;  hail  also  reaches  the 
ground  with  a  very  great  velocity  :  hence  we  may  conclude,  that 
it  is  formed  in  very  rare  air,  and  in  a  high  region  of  the  atmos- 
phere. The  decomposition  of  organic  substances,  is  constantly 
giving  out  hydrogen  gas,  and  this,  from  its  specific  levity,  rises 
to  the  higher  regions  of  the  atmosphere  ;  hence,  as  no  gas  can  re- 
main long  over  another  unmixed,  it  mingles  with  atmospheric  air, 
and  becomes  susceptible  of  hoing  inflamed  by  electricity.  Should 
it  be  thus  acted  upon,  it  forms  water,  will  be  condensed  into  a 
space  much  less  than  it  formerly  occupied,  and  would  leave  a  va- 
cuum, did  not  the  adjacent  portions  of  air  rush  in  to  fill  the  void. 
The  sudden  rarefaction  of  this  air  will  produce  an  intense  cold  ; 
the  newly  formed  water  will  be  frozen,  and  under  circumstances 
that  will  cause  it  to  granulate  ;  descending  from  a  lofty  region,  it 
will  have  great  velocity ;  formed  from  hydrogen  gas,  and  by  the 
electric  discharge,  it  will  occur  most  frequently  during  the  sum- 
mer months,  and  accompany  lightning. 

499.  It  was  shown  in  the  preceding  chapter,  that  the  earth  retains 
a  constant  mean  temperature,  under  the  joint  action  of  solar  and 
terrestrial  radiation  ;  but  that  the  rate  of  these  is  unequal,  not  only 
at  different  seasons,  but  from  hour  to  hour  ;  the  former  ceases  alto- 


Book   VL~\  THE  ATMOSPHERE.  507 

gether  at  the  setting  of  the  sun,  while  the  latter  continues  for  a 
time  undiminished.  Hence  the  surface  of  the  earth  cools  rapidly 
after  sunset,  and  may  speedily  reach  the  dew-point  of  the  air  in 
contact  with  it.  So  soon  as  this  is  the  case,  moisture  begins  to 
be  precipitated,  and  a  cloud  is  formed,  the  descent  of  the  water  of 
which  this  is  composed,  forms  the  deposit,  that  we  call  Dew.  The 
cooling  will  be  propagated  slowly  upwards,  and  the  cloud  will  ap- 
pear to  rise  ;  notwithstanding  which,  the  moisture  of  which  it  is 
composed,  actually  falls.  After  some  hours,  the  earth  and  air 
will  assume  the  same  temperature,  and  the  cloud  will  disappear. 

The  first  morning  rays  of  the  sun,  passing  horizontally  through 
the  air,  will  heat  it,  long  before  their  influence  can  be  felt  upon 
the  ground.  The  air  will  therefore  acquire  a  greater  capacity  for 
moisture  :  if  there  be  any  water  in  the  vicinity,  vapour  will  rise, 
and  propagate  itself  through  the  mass  ;  but  as  the  ground  still  re- 
mains colder,  a  new  precipitation  will  ensue  ;  thus  dew  will  again 
be  formed,  and  moisture  occur  in  the  morning. 

500.  When  the  surface  of  the  ground,  or  of  any  other  substance, 
is  cooled  by  radiation  to  the  temperature  of  32°,  the  dew  is  frozen, 
and  takes  the  form  of  white  or  hoarfrost;  this  may  often  be  de- 
posited, when  the  temperature  both  of  the  air  and  of  the  ground 
at  a  very  small  depth,  is  above  that  of  freezing. 

501.  When  clouds  exist  in  the  atmosphere,  the  radiation  is  im- 
peded, and  dew  will  not  be  formed.     Thus  a  want  of  dew  is 
usually  a  prognostic  of  rain. 

When  the  air  is  still,  dew  is  most  copious,  and  thus  it  falls  in 
greatest  abundance  in  sheltered  situations,  and  frosts  will  continue 
later  in  the  spring,  and  begin  earlier  in  autumn,  in  vallies  than  on 
the  open  hills  in  the  vicinity. 

The  motion  of  air  mixes  the  portion  cooled  by  contact  with  the 
earth,  with  that  which  is  not,  and  brings  new  masses  into  contact ; 
hence,  although  the  loss  of  heat  by  radiation,  may  be  as  great  or 
even  greater,  the  ground  will  receive  heat  from  the  air,  and  the 
change  of  temperature  will  be  less.  In  conformity,  heavy  dews 
do  not  fall  during  the  prevalence  of  high  winds,  and  hoar  frosts 
rarely  occur  while  they  blow. 

Surfaces  that  radiate  well,  will  be  most  cooled,  and  will  in  con- 
sequence receive  the  greatest  quantity  of  dew  ;  and  thus  of  land 
frequently  tilled,  and  that  which  is  left  undisturbed,  the  latter  will 
derive  most  moisture  from  the  atmosphere  in  this  form. 

502.  In  the  prosecution  of  this  subject,  we  have  in  some  degree 
trespassed  upon  the  limits  of  pure  physical  science  :  this  was  how- 
ever necessary,  in  order  to  give  a  correct  view  of  the  phenomena, 
but  the  discussion  is  incomplete  from  the  propriety  of  confining 
ourselves  as  closely  as  possible  to  what  is  strictly  mechanical. 


508  CONCLUSION.  [Book  VI. 

So  also  in  a  variety  of  other  cases,  it  has  been  judged  expe- 
dient to  extend  our  investigations  into  collateral  branches  of 
knowledge,  while  subjects  of  no  small  extent,  and  more  imme- 
diately connected  with  Mechanics,  have  been  passed  over.  This 
extension  on  the  one  hand,  and  omission  on  the  other,  have  been 
determined  by  a  view  to  the  practical  applications  of  our  subject. 
These  applications  were  originally  intended  to  have  formed  a 
part  of  the  work,  and  by  means  of  them  its  true  scope  and  ob- 
jects would  have  become  apparent.  In  its  present  form,  how- 
ever, the  author  trusts  it  will  be  found  an  useful  introduction  to 
the  study  of  a  most  important  and  interesting  branch  of  science, 
whether  it  be  considered  in  its  immediate  connexion  with  the 
arts,  or  in  its  bearing  upon  knowledge  of  a  more  elevated  char- 
acter. 


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